Stochastic Analysis and Mathematical Finance - Institut for ...

Stochastic Analysisand Mathematical Financewith applications of the Malliavin calculusto the calculation of risk numbersAlexander SokolSpeciale for cand.scient graden i matematikInstitut for Matematiske FagKøbenhavns UniversitetThesis for the master degree in mathematicsDepartment of Mathematical SciencesUniversity of CopenhagenDate of Submission: September 8, 2008Advisor: Rolf PoulsenCo-advisors: Bo Markussen & Martin JacobsenDepartment of Mathematical SciencesUniversity of Copenhagen

Contents1 Introduction 52 Preliminaries 112.1 The Usual Conditions and Brownian motion . . . . . . . . . . . . . . . . 112.2 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Stopping times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 The Itô Integral 393.1 The space of elementary processes . . . . . . . . . . . . . . . . . . . . . 403.2 Integration with respect to Brownian motion . . . . . . . . . . . . . . . 513.3 Integration with respect to integrators in M L W . . . . . . . . . . . . . . 563.4 Integration with respect to standard processes . . . . . . . . . . . . . . . 673.5 The Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.6 Itô’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.7 Itô’s Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . 923.8 Girsanov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.9 Why The Usual Conditions? . . . . . . . . . . . . . . . . . . . . . . . . . 1173.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214 The Malliavin Calculus 1274.1 The spaces S and L p (Π) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2 The Malliavin Derivative on S and D 1,p . . . . . . . . . . . . . . . . . . 1344.3 The Malliavin Derivative on D 1,2 . . . . . . . . . . . . . . . . . . . . . . 1424.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) . . . . . . . . . . . . . . . . 1544.5 Further theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1744.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Chapter 1IntroductionThis thesis is about stochastic integration, the Malliavin calculus and mathematicalfinance. The final form of the thesis has developed very gradually, and was certainlynot planned in detail from the beginning.A short history of the development of the thesis. I started out with a vagueidea about developing the stochastic integral with respect to what in Øksendal (2005)is known as an Itô process, only to a higher level of rigor than seen in, say, Øksendal(2005) or Steele (2000), and in a self-contained manner, and furthermore showingthat this theory rigourously could be used to obtain the fundamental results of mathematicalfinance. The novelty of this is that ordinary accounts of this type almostalways either has some heuristic points or apply more advanced results, such as theLévy characterisation theorem, which are left unproved in the context of the account.I also wanted the thesis to contain some practical aspects, and I knew that the calculationof risk numbers is a very important issue for practicioners. I had heard about theresults of Fournié et al. (1999), concisely introduced by Benhamou (2001), applyingthe Malliavin calculus to the problem of calculating risk numbers such as the deltaor gamma. When investigating this, I obtained two conclusions. First, the litteraturepertaining the Malliavin calculus is rather difficult, with proofs containing very littledetail. Second, the numerical investigations of the applications of the Malliavin calculusto the calculation of risk numbers have mostly been constrained to the contextof the Black-Scholes model. My idea was to attempt to amend these two issues, first

6 Introductionoff by developing the fundamentals of the Malliavin calculus in higher detail, and secondlyby applying the results to a more advanced model than the Black-Scholes model,namely the Heston model.In the first few months of work, I developed the hope that I could find the timeand ability not only to construct the stochastic integral for Brownian motion andcontinuous finite variation procesess, but also consider discontinuous finite variationprocesses. Furthermore, I expected to be able to develop the entire fundament ofthe Malliavin calculus, including the Skorohod integral, the result on the Malliavinderivative of the solution to a stochastic differential equation and detailed proofs ofthe results of Fournié et al. (1999). I also vainly expected to find time to considerPDE methods for pricing in finance and to investigate calculation of risk numbers forpath-dependent options. In retrospect, all this was very optimistic, and I have endedup restricting myself to stochastic integration for continuous processes, discussing onlythe fundamental results of the Malliavin derivative operator and not even going as faras to introduce the Skorohod integral. This means that the theory of the Malliavincalculus presented here is not even strong enough to be used to prove the results ofFournié et al. (1999). On the other hand, it is indisputably very rigorous and detailed,and thus constitutes a firm foundation for further theory. Finally, I also had to forgetabout the idea of PDE methods and path-dependent options in finance.The final content of the thesis, then, is this: A development of the stochastic integralfor Brownian motion and continuous finite variation processes, applied to set up somesimple financial models and to produce sufficient conditions for the absence of arbitrage.The Malliavin deriative operator is also developed, but it is necessary to referto the litterature for the results applicable to finance. Some Monte Carlo methods forevaluating expectations and sensitivities are reviewed, and finally we apply all of thisto pricing and calculating risk numbers in the Black-Scholes and Heston models.It can be sometimes difficult to distinguish between well-known and original work.Many of the results in this thesis are well-known, but formulated in a considerablydifferent manner, often opting for the most pedagogical approach available. There arealso many minor original results. The highlights of the original contributions in thethesis are these:• In Theorem 3.8.3, a proof of the Girsanov theorem not invoking the Lévy characterisationtheorem, building on the ideas presented in Steele (2000).

7• In Theorem 4.3.5, an extension of the chain rule of the Malliavin calculus, removingthe condition of boundedness of the partial derivatives.• In Section 5.6, identification of an erroneous expression of the Malliavin estimatorin Benhamou (2001) for the Heston model and correction of this.Other significant results are of course all of the numerical results for the Black-Scholesand Heston models in Section 5.5 and Section 5.6. Also, the proof of agreement of theintegrals in Theorem 3.8.4 is new. Furthermore, in Appendix C there can be found anew proof of the existence of the quadratic variation for continuous local martingalesnot using stochastic integration at all. Also, a version of Urysohn’s Lemma is provedyielding an expression for the Lipschitz constant of the bump function.Structure of the thesis. Overall, the thesis first develops the stochastic integral,secondly develops some of the basic results of the Malliavin calculus and finally appliesall of this to mathematical finance.In Chapter 2, we develop some preliminary results which will be essential to the restof the thesis. We review some results on Brownian motion and the usual conditions,martingales and the like. Furthermore, we develop a general localisation concept whichwill be essential to the smooth development of the stochastic integral.Chapter 3 develops the Itô integral. We begin with some results on the space ofelementary processes, rigorously proving that the usual definition of the stochasticintegral is independent of the representation of the elementary process. Furthermore,we prove a general density result for elementary processes. These preparations makethe development of the integral for Brownian motion very easy. We then extendthe integral to integrate with respect to integral proceses and continuous processesof finite variation. After having constructed the stochastic integral in the level ofgenerality necessary, we define the quadratic variation and use it to prove Itô’s formula,Itô’s representation theorem and Girsanov’s theorem. We also prove Kazamaki’s andNovikov’s conditions for an exponental martingale to be a true martingale. Finally,we discuss the pros and cons of the usual conditions.We then proceed to Chapter 4, containing the basic results on the Malliavin calculus.We define the Malliavin derivative slightly differently than in Nualart (2006), initiallyusing coordinates of the Brownian motion instead of stochastic integrals with deterministicintegrands. We then extend the operator by the usual closedness argument.

8 IntroductionNext, we prove the chain rule and the integration-by-parts formula. We use the generalizedchain rule to show that our definition of the Malliavin derivative coincides withthat in Nualart (2006). Finally, we develop the Hilbert space theory of the Malliavinderivative and use it to obtain a chain rule for Lipschitz transformations.Chapter 5 contains the applications to mathematical finance. We first define a simpleclass of financial market models and develop sufficent criteria for the absence ofarbitrage, and we discuss what a proper price for a contingent claim is. After this,we take a quick detour to introduce some fundamental concepts from the theory ofSDEs, and we go through some Monte Carlo methods for evaulation of expectationsand sensitivities. Finally, we begin the numerical experiments. We first consider theBlack-Scholes model, where we evaluate call prices, call deltas and digital deltas. Thepurpose of this is mainly a preliminary evaluation of the methods in a simple context,allowing us to gain an idea of their effectiveness and applicability. After this, we tryour luck at the Heston model. We prove absence of arbitrage building on some ideasfound in Cheridito et al. (2005). We then consider some discretisation schemes forsimulating from the Heston model, and apply these to calculation of the digital delta.We compare different methods and conclude that the localised Malliavin method isthe superior one. Finally, we outline the difference between our Malliavin estimatorand the one found in Benhamou (2001), illustrating the resulting discrepancy.Finally, in Chapter 6, we discuss our results and opportunities for further work.The appendices also contain several important results, with original proofs. AppendixA mostly contains well-known results, but the sections on Hermite polynomials andHilbert spaces contain proofs of results which are hard to find in the litterature. AppendixB contains basic results from measure theory, but also has in Section B.3 anoriginal result on convergence of normal distributions which is essential to our developmentof the Hilbert space theory of the Malliavin calculus. Furthermore, in AppendixC, we prove some novel results which ultimately turned out not to be necessary for themain parts of the thesis: An existence proof for the quadratic variation and a Lipschitzversion of Urysohn’s lemma.Prerequisites. The thesis is written with a reader in mind who has a good grasp ofthe fundamentals of real analysis, measure theory, probability theory such as can befound in Carothers (2000), Hansen (2004a), Dudley (2002), Jacobsen (2003) andRogers & Williams (2000a). Some Hilbert space theory and functional analysis is alsoapplied, for introductions to this, see for example Hansen (2006) and Meise & Vogt

9(1997).Final words. I would like to thank by advisor Rolf Poulsen and by co-advisors BoMarkussen and Martin Jacobsen for their support in writing this thesis. Furthermore,I would like to give a special warm thanks to Martin Jacobsen and Ernst Hansen forteaching me measure theory and probability theory and spending so much time alwaysanswering mails and discussing whatever mathematical problems I have had at hand.

10 Introduction

Chapter 2PreliminariesThis chapter contains results which are fundamental to the theory that follows, inparticular the stochastic calculus, which relies heavily on the machinery of stoppingtimes, martingales and localisation. Many of the basic results in this chapter areassumed to be well-known, and we therefore often refer to the litterature for proofsof the results. Other results, such as the results on progressive measurability and theBrownian filtration, are more esoteric, and we give full proofs.The first chapter of Karatzas & Shreve (1988) and the second chapter of Rogers &Williams (2000a) are excellent sources for most of the results in this section.2.1 The Usual Conditions and Brownian motionLet (Ω, F, P, F t ) be a filtered probability space. Recall that the usual conditions for afiltered probability space are the conditions:1. The σ-algebra F is complete.2. The filtration is right-continuous, F t = ∩ s>t F s .3. For t ≥ 0, F t contains all null sets in F.

12 PreliminariesIn every section of this thesis except this one, we are going to assume the usualconditions. We are also going to work with Brownian motion throughout this thesis.We therefore need to spend some time making sure that we can make our Brownianmotion work together with the usual conditions. This is the subject matter of thissection.The reasons for assuming the usual conditions and ways to avoid assuming the usualconditions will be discussed in Section 3.9. Many of our results also hold without theusual conditions, but to avoid clutter, we will consistently assume the usual conditionsthroughout the thesis.Our first aim is to understand how to augment a filtered probability space so that itfulfills the usual conditions. In the following, N denotes the null sets of F. Before wecan construct the desired augmentation of a filtered probability space, we need a fewlemmas.Lemma 2.1.1. Letting G = σ(F, N ), it holds thatG = {F ∪ N|F ∈ F, N ∈ N },and P can be uniquely extended from F to a probability measure P ′ on G by puttingP ′ (F ∪ N) = P (F ). The space (Ω, G, P ′ ) is called the completion of (Ω, F, P ).Proof. We first prove the equality for G. Define H = {F ∪ N|F ∈ F, N ∈ N }. It isclear that H ⊆ G. We need to prove the opposite inclusion. To do so, we prove directlythat H is a σ-algebra containing F and N , the inclusion follows from this. It is clearthat Ω ∈ H. If H ∈ H with H = F ∪ N, we obtain, with B ∈ F such that P (B) = 0and N ⊆ B,H c = (F ∪ N) c= (B c ∩ (F ∪ N) c ) ∪ (B ∩ (F ∪ N) c )= (B ∪ F ∪ N) c ∪ (B ∩ (F ∪ N) c )= (B ∪ F ) c ∪ (B ∩ (F ∪ N) c ),so since (B ∪ F ) c ∈ F and B ∩ (F ∪ N) c ∈ N , we find H c ∈ H. If (H n ) ⊆ H withH n = F n ∪ N n , we find(n⋃∞) (⋃∞)⋃H n = F n ∪ N n ,n=1n=1n=1

2.1 The Usual Conditions and Brownian motion 13showing ∪ ∞ n=1H n ∈ H. We have now proven that H is a σ-algebra. Since it containsF and N , we conclude G ⊆ H.It remains to show that P can be uniquely extended from F to G. We begin by provingthat the proposed extension is well-defined. Let G ∈ G with two decompositionsG = F 1 ∪ N 1 and G = F 2 ∪ N 2 . We then have G c = F2 c ∩ N2. c Thus, if ω ∈ F2 c andω ∈ G, then ω ∈ N 2 , showing F2 c ⊆ G c ∪ N 2 and thereforeF 1 ∩ F c 2 ⊆ F 1 ∩ (G c ∪ N 2 ) = F 1 ∩ N 2 ⊆ N 2 ,and analogously F 2 ∩F c 1 ⊆ N 1 . We can therefore conclude P (F 1 ∩F c 2 ) = P (F 2 ∩F c 1 ) = 0and as a consequence,P (F 1 ) = P (F 1 ∩ F 2 ) + P (F 1 ∩ F c 2 ) = P (F 2 ∩ F 1 ) + P (F 2 ∩ F c 1 ) = P (F 2 ).This means that putting P ′ (G) = P (F ) for G = F ∪ N is a definition independent ofthe representation of G, therefore well-defined. That P ′ is a probability measure on Gextending P is obvious.In the following, we let (Ω, G, P ) be the completion of (Ω, F, P ). In particular, we usethe same symbol for the measure on F and its completion. This will not cause anyproblems.Lemma 2.1.2. Let H be a sub-σ-algebra of G. Thenσ(H, N ) = {G ∈ G|G∆H ∈ N for some H ∈ H},where ∆ is the symmetric difference, G∆H = (G \ H) ∪ (H \ G).Proof. Define K = {G ∈ G|G∆H ∈ N for some H ∈ H}. We begin by arguing thatK ⊆ σ(H, N ). Let G ∈ K and let H ∈ H be such that G∆H ∈ N . We then findG = (H ∩ G) ∪ (G \ H)= (H ∩ (H ∩ G c ) c ) ∪ (G \ H)= (H ∩ (H \ G) c ) ∪ (G \ H).Since P (G∆H) = 0, H \ G and G \ H are both null sets, and we conclude thatG ∈ σ(H, N ). Thus K ⊆ σ(H, N ).To show the other inclusion, we will show that K is a σ-algebra containing H andN . If G ∈ H, we have G∆G = ∅ ∈ N and therefore G ∈ K. If N ∈ N , we have

18 PreliminariesThe conclusion follows.Theorem 2.1.8 is the result that will allow us to assume the usual conditions in whatfollows. Next, we prove Blumenthal’s zero-one law.Lemma 2.1.9. It holds that F t+ = σ(F t , N t+ ), where N t+ are the null sets of F t+ .Proof. It is clear that σ(F t , N t+ ) ⊆ F t+ , so it will suffice to show the other inclusion.Let A ∈ F t+ , and put ξ = 1 A − E(1 A |F t ). Our first step will be to prove that ξis almost surely zero, and to do so, we first prove E(ξ1 B ) = 0 for B ∈ F ∞ , whereF ∞ = σ(∪ t≥0 F t ).To this end, note that with U t = σ(W t+s − W t ) s≥0 , F ∞ = σ(F t , U t ). The sets of theform C ∩ D, where C ∈ F t and D ∈ U t , are then a generating system for F ∞ , stableunder intersections. Using Lemma 2.1.6, we see that U t and F t+ are independent.Since ξ is F t+ measurable, we conclude, with C ∈ F t and D ∈ U t ,E(ξ1 C∩D ) = P (D)E(ξ1 C )= P (D)E((1 A − E(1 A |F t ))1 C ).Now, by the definition of conditional expectation, EE(1 A |F t )1 C ) = E1 A 1 C , and weconclude E(ξ1 C∩D ) = 0. Then, by the Dynkin Lemma, E(ξ1 B ) = 0 for all B ∈ F ∞ ,and therefore ξ is zero almost surely. Since ξ is F t+ measurable, this means that thereexists N ∈ F t+ with P (N) = 0 such that ξ = 0 on N c , and therefore1 A = 1 A 1 N + 1 A 1 N c= 1 A∩N + ξ1 N c + E(1 A |F t )1 N c= 1 A∩N + E(1 A |F t )1 N c.Because A ∩ N ∈ N t+ , the right-hand side is measurable with respect to σ(F t , N t+ )and therefore A ∈ σ(F t , N t+ ).Lemma 2.1.10 (Blumenthal’s 0-1 law). For any A ∈ G 0 , P (A) is either zero orone.Proof. Let H be the sets of G 0 where the lemma holds. It is clear that H is a Dynkinsystem. Using Lemma 2.1.9, we obtainG 0 = σ(F 0+ , N ) = σ(F 0 , N 0+ , N ) = σ(F 0 , N ).

FEDERAZIONE ITALIANA PALLACANESTROCOMITATO ITALIANO ARBITRIIn tutte le situazioni, nel corso della gara, in cui gli arbitri segnalano unsalto a due si eseguirà una rimessa dalle linee perimetrali secondo laregola del possesso alternato.Si evidenzia che il principio del possesso alternato si applica solo nei casisopraindicati, mentre tutte le “normali” rimesse laterali seguono ledisposizioni regolamentari.Nel processo del possesso alternato la palla diventa viva con una rimessain gioco dalle linee perimetrali, anziché con un salto a due.Nel corso della gara, quando si verifica una situazione di salto a due(fischio arbitrale seguito dal segnale n° 24 braccia alte e pollici rivoltiverso l’alto,seguiti dall’indicazione della direzione della freccia ), il giocoriprenderà procedendo con il metodo del possesso alternato.Il diritto di una squadra al possesso alternato sarà sempre indicato da unaPALETTA – FRECCIA visibile ( manuale od elettronica ) posta sul tavolodegli Ufficiali di Campo, che punterà in direzione del canestro avversario.*dal regolamento : IL CANESTRO AVVERSARIO È QUELLO IN CUI LASQUADRA SEGNA I PUNTI VALIDI *Responsabile del posizionamento della paletta-freccia è il segnapunti.A seguito del salto a due effettuato all’inizio del 1° periodo il segnapuntiindividuerà la squadra che ha acquisito il primo possesso legale dellapalla e posizionerà la paletta-freccia nella direzione del canestro dellasquadra stessa ( quello in cui attacca la squadra avversaria)Nelle eventuali successive situazioni di salto a due le squadre sialterneranno nell’effettuare la rimessa dalle linee perimetrali del campo.Il verso della paletta/freccia, indicante il nuovo diritto di rimessa laterale,verrà cambiato nel momento in cui, dopo che l’arbitro ha messo la palla adisposizione del giocatore, questa è stata toccata o ha toccato ungiocatore in campo.Nella situazione in cui la squadra che effettua la rimessa da fuori campoper possesso alternato, commetta una violazione alle norme stabilite perquesto caso ( spostamento dal punto di rimessa o utilizzo di più di 5secondi per la rimessa stessa) perde l’opportunità del diritto alla rimessaalternata.IL SEGNAPUNTI CAMBIERÀ LA DIREZIONE DELLA PALETTA/FRECCIA.Nella situazione in cui, nel corso di una rimessa per possessoalternato,venga commesso un fallo da una delle due squadre, la squadranon perderà il diritto di rimessa, l’arbitro procederà alla gestione dellasanzione e, ALLA RIPRESA DEL GIOCO, IL SEGNAPUNTI NONCAMBIERÀ IL VERSO DELLA PALETTA/FRECCIA.Nella situazione di salto a due iniziale, con possesso illegale della palla,l’arbitro fischierà la violazione e farà rimettere dalle linee perimetrali allasquadra avente diritto.Nel momento preciso in cui la palla verrà toccata in campo,Pagina 20 di 20SETTEMBRE 2006

2.2 Stochastic processes 21Lemma 2.2.2. Let Σ π be the familiy of sets A ∈ B[0, ∞) ⊗ F such that A satisfiesA ∩ [0, t] × Ω ∈ B[0, t] ⊗ F t for all t ∈ [0, ∞). Then Σ π is a σ-algebra, and a processX is progressively measurable if and only if it is Σ π -measurable.Proof. That Σ π is a σ-algebra is clear. The equality(X ∈ A) ∩ [0, t] × Ω = (X |[0,t]×Ω ∈ A)shows that progressive measurability is equivalent to Σ π measurability.Lemma 2.2.1 and Lemma 2.2.2 show that the concepts of being measurable andadapted and of being progressive correspond to conventional measurability propertieswith respect to the σ-algebras Σ A and Σ π , respectively. These results will enableus to apply the usual machinery of measure theory when dealing when these measurabilityconcepts. The next results demonstrate some of the interplay between theregularity properties of stochastic processes. In particular, Lemma 2.2.3 shows thatΣ π ⊆ Σ A .Lemma 2.2.3. Let X be progressive. Then X is measurable and adapted.Proof. That X is measurable follows from Lemma 2.2.2. To show that X is adapted,note that when X is progressive, X |[0,t]×Ω is B[0, t] ⊗ F t -measurable, and thereforeω ↦→ X(t, ω) is F t -measurable.Lemma 2.2.4. Let X be adapted and continuous. Then X is progressively measurable.Proof. Define X n (t) = X( 12 n [2 n t]). We can then writeX n (t) =∞∑k=0( ) kX2 n 1 [k2 n , k+1)(t).2 nSince X is adapted, X( k2) is F n k measurable. Therefore, each term in the sum is2n progressive. Since X n converges pointwise to X by the continuity of X, we concludethat X is progressive as the limit of progressive processes.Comment 2.2.5 In the proof of the lemma, we actually implicitly employed the resultfrom Lemma 2.2.2 that progressive measurability is equivalent to measurability withrespect to a σ-algebra Σ π , in the sense that we know that ordinary measurability

22 Preliminariesis preserved by pointwise limits. Had we not known that progressive measurability ismeasurability with respect to Σ π , we would have to prove manually that it is preservedby pointwise convergence. Note that the result would also hold if continuity wasweakened to left-continuity or right-continuity.◦Lemma 2.2.6. Let X and Y be measurable processes. If X and Y are versions, theset {(t, ω) ∈ [0, ∞) × Ω|X t (ω) ≠ Y t (ω)} is a λ ⊗ P null set.Proof. First note that A = {t, ω) ∈ [0, ∞) × Ω|X t (ω) ≠ Y t (ω)} is B[0, ∞) ⊗ F measurablesince X and Y both are measurable. By the Tonelli Theorem, thenshowing the claim.(λ ⊗ P )(A) =∫ ∞0P (X t ≠ Y t ) dt = 0,Comment 2.2.7 Note that the other implication does not hold: We can only concludethat P (X t ≠ Y t ) = 0 almost surely if {(t, ω) ∈ [0, ∞) × Ω|X t ≠ Y t } is a λ ⊗ P null set.Note also that it is crucial that both X and Y are measurable in the above, otherwisethe set {(t, ω) ∈ [0, ∞) × Ω|X t (ω) ≠ Y t (ω)} would not be B[0, ∞) ⊗ F measurable. ◦2.3 Stopping timesThis section contains some basic results on stopping times. A stopping time is astochastic variable τ : Ω → [0, ∞] such that (τ ≤ t) ∈ F t for any t ≥ 0. We say thatτ is finite if τ maps into [0, ∞). We say that τ is bounded if τ maps into a boundedsubset of [0, ∞). If X is a stochastic process and τ is a stopping time, we denote by X τthe process Xt τ = X τ∧t and call X τ the process stopped at τ. We denote by X[0, τ]the process X[0, τ] t = X t 1 [0,τ] (t) and call X[0, τ] the process zero-stopped at τ.Furthermore, we define the stopping-time σ-algebra F τ of events determined at τ byF τ = {A ∈ F|A ∩ (τ ≤ t) ∈ F t for all t ≥ 0}. Clearly, F τ is a σ-algebra. Ourfirst goal is to develop some basic results on stopping times and their interplay withstopping-time σ-algebras. By B 0 , we denote the elements A ∈ B with 0 /∈ A.Lemma 2.3.1. If τ and σ are stopping times, then so is τ ∧ σ, τ ∨ σ and τ + σ.Furthermore, if σ ≤ τ, then F σ ⊆ F τ .

2.3 Stopping times 23Proof. See Karatzas & Shreve (1988), Lemma 1.2.9 and Lemma 1.2.15.Lemma 2.3.2. Let τ be a stopping time. Then (τ < t) ∈ F t for all t ≥ 0.Proof. We have (τ < t) = ∪ n≥1(τ ≤ t −1n)∈ Ft , since (τ ≤ t − 1 n ) ∈ F t− 1 n ⊆ F t.Lemma 2.3.3. Let X be progressively measurable, and let τ be a stopping time. ThenX τ is F τ measurable and X τ is progressively measurable.Proof. See Karatzas & Shreve (1988), Proposition 1.2.18.Lemma 2.3.4. For any stopping time τ, the process (t, ω) ↦→ 1 [0,τ(ω)] (t) is progressive.In particular, if X is Σ π measurable, then X[0, τ] is Σ π measurable as well.Proof. Let t ≥ 0. Our fundamental observation is{(s, ω) ∈ [0, t] × Ω|1 [0,τ(ω)] (s) = 0} = {(s, ω) ∈ [0, t] × Ω|τ(ω) < s}.We need to prove that the latter is B[0, t] ⊗ F t measurable. Since 1 [0,τ] only takesthe values 0 and 1, this is sufficient to prove progressive measurability. To this end,define the mapping f : [0, t] × (τ ≤ t) → [0, t] 2 by f(s, ω) = (s, τ(ω)). Let τ t be therestriction of τ to (τ ≤ t). We then have (τ t ≤ s) ∈ F s ⊆ F t for any s ≤ t. Sincethe sets of the form [0, s], s ≤ t, form a generating family for B[0, t], we conclude thatτ t is F t − B[0, t] measurable. Therefore, f is B[0, t] ⊗ F t − B[0, t] 2 measurable. WithA = {(x, y) ∈ [0, t] 2 |x > y}, A ∈ B[0, t] 2 and we obtain{(s, ω) ∈ [0, t] × Ω|τ(ω) < s} = {(s, ω) ∈ [0, t] × Ω|f(s, ω) ∈ A}= f −1 (A),which is B[0, t]⊗F t measurable, as desired. The remaining claims of the lemma followsimmediately.Lemma 2.3.5. Let X be a continuous and adapted process, and let F be closed. Thenthe first entrance time τ = inf{t ≥ 0|X t ∈ F } is a stopping time.Proof. See Rogers & Williams (2000a), Lemma 74.2.Lemma 2.3.6. Let τ k , k ≤ n be a family of stopping times. Then the i’th orderedvalue of (τ k ) k≤n is also a stopping time.

24 PreliminariesComment 2.3.7 Note that it does not matter how we order the stopping times incase of ties, since we are only interested in the ordered values of the stopping times,not the actual ordering.◦Proof. Let t ≥ 0, and let τ (i) be the i’th ordered value of (τ k ) k≤n . Then τ (i) ≤ tprecisely if i or more of the stopping times are less than t. That is,⋃(τ (i) ≤ t) =(τ k ≤ t, k ∈ I) ∩ (τ k > t, k /∈ I) ∈ F t .I⊆{1,...,n},|I|≥iLemma 2.3.8. Let τ and σ be stopping times. Assume that Z ∈ F σ . Then it holdsthat both Z1 (σ

2.4 Martingales 25Since Z ∈ F σ , we find (Z ∈ B) ∩ (σ ≤ t) ∈ F t . And since we know (τ < σ) ∈ F τ∧σ ,(σ ≤ τ) = (τ < σ) c ∈ F σ∧τ , so (σ ≤ τ) ∩ (σ ∧ τ ≤ t) ∈ F t . This demonstrates(Z1 (σ≤τ) ∈ B) ∩ (σ ∧ τ ≤ t) ∈ F t , as desired.2.4 MartingalesIn this section, we review the basic theory of martingales. As before, we work in afiltered probability space (Ω, F, F t , P ) satisfying the usual conditions. If M is someprocess, we put M ∗ t = sup s≤t |M s | and M ∗ = M ∗ ∞. A process M is said to be amartingale if M s is adapted and E(M t |F s ) = M s whenever s ≤ t. We say that M issquare-integrable if M is bounded in L 2 . As described in Appendix B.5, we say thatM is uniformly integrable iflim sup E|M t |1 (|Mt |>x) = 0.x→∞t≥0In this case, M is also bounded in L 1 by Lemma B.5.2. By Lemma B.5.1, if M t isbounded in L p for any p > 1, it is uniformly integrable. We denote the space ofmartingales by M. The space of martingales stating at zero is denoted M 0 .We will mostly be concerned with continuous martingales. However, at some pointswe will be interested in cadlag martingales as well. The following result shows that wealways can find a cadlag version of any martingale.Theorem 2.4.1. Let M be a martingale. Then M has a cadlag version.Proof. This follows from Theorem II.67.7 of Rogers & Williams (2000a).Next, we go on to state some of the classic results of martingale theory.Theorem 2.4.2 (Doob’s maximal inequality). Let M be a nonnegative cadlagsubmartingale. For c > 0 and t ≥ 0, cP (Mt∗ ≥ c) ≤ EM t 1 (M ∗t ≥c).Proof. See Rogers & Williams (2000a), Theorem 70.1.Theorem 2.4.3 (Martingale Convergence Theorem). Let M be a continuoussubmartingale. If M + is bounded in L 1 , then M is almost surely convergent to an

26 Preliminariesintegrable variable. If M is also uniformly integrable, the limit M ∞ exists in L 1 aswell, and E(M ∞ |F t ) = M t . In this case, we say that M ∞ closes the martingale.Proof. The first part follows from Theorem 1.3.15 in Karatzas & Shreve (1988), andthe rest follows from Theorem II.69.2 in Rogers & Williams (2000a).Corollary 2.4.4. Let M be a continuous martingale. M is closed by a variable M ∞if and only if M is uniformly integrable, and in this case M t converges to M ∞ almostsurely and in L 1 .Proof. From Theorem 2.4.3, we know that if M is uniformly integrable, M is closedby some variable and we have convergence almost surely and in L 1 . It will thereforesuffice to show the other implication. Assume that M is closed by a variable M ∞ , weneed to show that M is uniformly integrable. By Jensen’s inequality, we know that|M t | = |E(M ∞ |F t )| ≤ E(|M ∞ ||F t ). In particular, M is bounded in L 1 , and thereforeM t is convergent almost surely. Therefore, M ∗ is almost surely finite. We obtainE|M t |1 (|Mt|>x) ≤ EE(|M ∞ ||F t )1 (|Mt|>x)= E|M ∞ |1 (|Mt|>x)and since M ∗ is almost surely finite, we concludelim supx→∞supt≥0so M is uniformly integrable.≤E|M ∞ |1 (M ∗ >x),E|M t |1 (|Mt|>x) ≤ lim sup E|M ∞ |1 (M ∗ >x) = 0,x→∞Theorem 2.4.5 (Doob’s L p -inequality). Let M be a continuous martingale. Forany t ≥ 0 and p > 1, ‖M ∗ t ‖ p ≤ q‖M t ‖ p , where q is the dual exponent to p.Proof. See Rogers & Williams (2000a), Theorem 70.2.Lemma 2.4.6. If M is a martingale bounded in L 2 , then M is almost surely convergentand the limit M ∞ is square integrable. Furthermore, M ∞ closes the martingale.Proof. See Rogers & Williams (2000a), Theorem 70.2.

2.4 Martingales 27Theorem 2.4.7 (Optional Sampling). Let M be a continuous martingale. Letσ ≤ τ be stopping times. Then E(M τ |F σ ) = M σ if either M is uniformly integrableor σ and τ are bounded.Proof. Assume first that M is uniformly integrable. By 2.4.3, M is then convergentalmost surely and in L 1 with limit M ∞ , and M ∞ closes the martingale. Then Theorem1.3.22 of Karatzas & Shreve (1988) yields the result.Next assume that σ and τ are bounded, say by T . Then M τ = M T τ and M σ = M T σ .M T is a uniformly integrable martingale by Corollary 2.4.4, so we obtainE(M τ |F σ ) = E(M T τ |F σ ) = M T σ = M σ .Since any square-integrable martingale is uniformly integrable, optional sampling holdsfor square-integrable martingales for all stopping times.Definition 2.4.8. By cM 2 0, we denote the space of continuous martingales boundedin L 2 . We endow cM 2 0 with the the seminorm given by ‖M‖ cM 20= √ EM 2 ∞.It is clear that ‖ · ‖ cM 20is a well-defined seminorm.Lemma 2.4.9. Convergence in cM 2 0 is equivalent to uniform L 2 -convergence andimplies almost sure uniform convergence along a subsequence. Also, ‖M − N‖ cM 20= 0if and only if M and N are indistinguishable.Proof. From Doob’s L 2 -inequality, Lemma 2.4.5, we have for any M, N ∈ cM 2 0 that‖M − N‖ cM 20≤ ‖(M − N) ∗ ‖ 2 ≤ 2‖M ∞ − N ∞ ‖ 2 = 2‖M − N‖ cM 20.Therefore, it is immediate that convergence in cM 2 0 is equivalent to uniform L 2 -convergence. Now assume that M n converges to M in cM 2 0. In particular, then(M n − M) ∗ converges to 0 in L 2 . Therefore, there is a subsequence converging almostsurely, which means that (M nk −M) ∗ converges almost surely to 0, and this is preciselythe almost sure uniform convergence proposed.To se that ‖M−N‖ cM 20= 0 if and only if M and N are indistinguishable, first note thatif M and N are indistinguishable, then M ∞ = lim M t = lim N t = N ∞ almost surely,

28 Preliminariesand therefore ‖M − N‖ cM 20= ‖M ∞ − N ∞ ‖ 2 = 0. Conversely, if ‖M − N‖ cM 20= 0then by Doob’s L 2 -inequality, ‖(M − N) ∗ ‖ 2 = 0, showing that (M − N) ∗ is almostsurely zero, and this implies indistinguishability.Lemma 2.4.10. The space cM 2 0 is complete.Proof. Let (M n ) be a cauchy sequence in cM 2 0. Then (M n ∞) is a cauchy sequencein L 2 (F ∞ ), therefore convergent to some M ∞ ∈ L 2 (F ∞ ). Now define M by puttingM t = E(M ∞ |F t ), it is clear from Jensen’s inequality that M is a martingale boundedin L 2 . Since‖(M n − M) ∗ ∞‖ 2 ≤ 2‖M n ∞ − M ∞ ‖ 2 ,we find that M n converges uniformly in L 2 to M. As in the proof of Lemma 2.4.9, wethen find that there is a subsequence such that M n kconverges almost surely uniformlyto M. In particular, since M n kis continuous, M is almost surely continuous. Sincewe have assumed the usual conditions, we can pick be a modification N of M whichis adapted and continuous for all sample paths. We then find that N ∈ cM 2 0 and M nconverges to N in cM 2 0, as desired.Lemma 2.4.11. Let M be a continuous martingale and let τ be a stopping time. Thenthe stopped process M τ is also a continuous martingale.Proof. Let 0 ≤ s ≤ t. We writeE(M τ t |F s ) = 1 (τ≤s) E(M τ t |F s ) + 1 (τ>s) E(M τ t |F s )and consider the two terms separately. Since 1 (τ≤s) is F s measurable, we find1 (τ≤s) E(M τ t |F s ) = E(M τ t 1 (τ≤s) |F s )= E(M τ s 1 (τ≤s) |F s )= M τ s 1 (τ≤s) ,since M τ t 1 (τ≤s) = M τ s 1 (τ≤s) , which is F s measurable. For the second term, we find1 (τ>s) E(M τ t |F s ) = E(1 (τ>s) M τ t |F s ), and for F ∈ F s we have F ∩ (τ > s) ∈ F τ∧s byLemma 2.3.8, yieldingE(1 F E(1 (τ>s) M τ t |F s )) = E(1 F 1 (τ>s) M τ t )= E(1 F 1 (τ>s) E(M τ t |F τ∧s ))= E(1 F 1 (τ>s) M τ s ),

2.4 Martingales 29by optional sampling, so that 1 (τ>s) E(M τ t |F s ) = 1 (τ>s) M τ s . All in all, we concludeE(M τ t |F s ) = M τ s .The following lemma will be useful in several of the calculations we are to make whendeveloping the stochastic integral.Lemma 2.4.12. Let M ∈ cM 2 0, and let (t 0 , . . . , t n ) be a partition of [s, t]. Then( n∣ )∑∣∣∣∣E (M tk − M tk−1 ) 2 F s = E((M t − M s ) 2 |F s )k=1Proof. We see thatE((M t − M s ) 2 |F s )⎛( n) ∣ ⎞ 2∑∣∣∣∣∣= E ⎝ M tk − M tk−1 F s⎠k=1n∑n∑= E((M tk − M tk−1 ) 2 |F s ) + E((M tk − M tk−1 )(M ti − M ti−1 )|F s ),k=1k≠iwhere, for k < i,E((M tk − M tk−1 )(M ti − M ti−1 )|F s )= E((M tk − M tk−1 )E(M ti − M ti−1 |F ti−1 )|F s )= 0.This shows the lemma.We end the section with a very useful criterion for determining when a process is amartingale or a uniformly integrable martingale.Lemma 2.4.13. Let M be a progressive process such that the limit exists almost surely.If EM τ = 0 for any bounded stopping time τ, M is a martingale. If EM τ = 0 for anystopping time τ, M is a uniformly integrable martingale.Proof. The statement that M is a uniformly integrable martingale if EM τ = 0 for anystopping time τ is Theorem II.77.6 of Rogers & Williams (2000a). If we only havethat EM τ = 0 for any bounded stopping time, M t is a uniformly integrable martingalefor any t ≥ 0 and therefore, M is a martingale.

30 Preliminaries2.5 LocalisationIn our construction of the stochastic integral, we will employ the technique of localisation,where we extend properties and definitions to processes where a given propertyonly holds locally in the sense that if we somehow cut off the process from a certainpoint onwards, then the property holds.We will mainly be using two localisation concepts: With H a family of stochasticprocesses, we say that a process is locally in H if there exists a sequence of stoppingtimes increasing to infinity such that X τn is in H and we say that a process is zerolocallyin H if there exists a sequence of stopping times increasing to infinity such thatX[0, τ n ] is in H for n ≥ 1. We also write that X is L-locally in H and that X isL 0 -locally in H, respectively.We will need a variety of results about these localisation procedures. To avoid havingto prove the same results twice, we define an abstract concept of localisation andprove the results for this abstract type of localisation. This also has the convenientside-effect of demonstrating what is the essential properties of localisation: The abilityto determine a process from its local properties, and the ability to paste together a newprocess from localised processes. Our definition of localisation is sufficiently strong tohave these properties, but also sufficiently broad to encompass the two concepts oflocalisation that we will need.In the following, we denote by SP the set of real stochastic processes on a filteredprobability space (Ω, F, P, F t ) satisfying the usual conditions.Definition 2.5.1. A localisation concept L is a family of mappings f τindexed by the set of stopping times, such that: SP → SP,1. For any stopping times τ and σ, f τ ◦ f σ = f τ∧σ .2. If f τ (X) = f τ (Y ), then X t (ω) = Y t (ω) whenever t ≤ τ(ω).The interpretation of the above is the following. The mappings f τ correspond tocutoff functions, such that f τ (X) is the restriction of X to {(t, ω)|t ≤ τ(ω)} in somesense. The first property ensures that the order of cutting off is irrelevant. The secondproperty clarifies that f τ (X) corresponds to cutting off X to {(t, ω)|t ≤ τ(ω)}. The twolocalisation concepts mentioned earlier is covered by our definition. Specifically, define

2.5 Localisation 31the localisation concept of stopping L as the family of mappings f τ (X) = X τ and thelocalisation concept of zero-stopping L 0 as the family of mappings f τ (X) = X[0, τ].Then L and L 0 are both localisation concepts according to Definition 2.5.1.Our abstract localisation concept does not cover more esoteric localisation conceptssuch as the concept of pre-stopping as found in Protter (2005), Chapter IV. If, however,we exchanged {(t, ω)|t ≤ τ(ω)} with {(t, ω)|t < τ(ω)}, it would cover pre-stoppingas well. This is unnecessary for our purposes, however, and we therefore omit thisextension.In the following, let L = (f τ ) be a localisation concept. Let H be a subset of SP.Definition 2.5.2. We say that a sequence τ n of stopping times is determining if τ nincreases to infinity. Let X be a stochastic process. We say that X is L-locally in Hif there is a determining sequence τ n such that f τn (X) ∈ H for all n ≥ 1. In this case,we say that τ n is a localising sequence for X. The set of elements L-locally in H isdenoted L(H).Note that in the above, we defined two different labels for sequences of stopping times.A sequence of stopping times (τ n ) is determining if it increases to infinity. The samesequence is localising for X to H if f τn (X) ∈ H for n ≥ 1. The interpretation of Definition2.5.2 is that when a process X is locally in H, we can cut off with the localisingsequence τ n and then the result f τn (X) will be in H. The reason for using determiningsequences is to ensure that the “localised” versions f τn (X) actually determine theelement, as the following result shows.Lemma 2.5.3. If τ n is a determining sequence and f τn (X) = f τn (Y ) for all n ≥ 1,then X and Y are equal.Proof. It follows that X t (ω) and Y t (ω) are equal whenever t ≤ τ n (ω). Since τ n tendsto infinity, we obtain X = Y .We will now introduce the concept of a stable subset and develop some basic resultsabout localisation and stability. When we have done this, we continue to the mainresults about localisation: The ability to paste localised processes together and theability to extend mappings H 1 → H 2 to mappings between localised versions of H 1and H 2 . In order to make the contents of the following easier to understand, we willcomment on the meaning of the results in the context of our staple example, that of

34 PreliminariesTheorem 2.5.11. Let L 1 = (f τ ) and L 2 = (g τ ) be two localisation concepts. Let H 1and H 2 be spaces of stochastic processes such that H 1 is L 1 -stable and H 2 is L 2 -stable.Let α : H 1 → H 2 be a mapping with g τ (α(X)) = α(f τ (X)). There exists an extensionα : L 1 (H 1 ) → L 2 (H 2 ), uniquely defined by the criterion that g τ (α(X)) = α(f τ (X)).Proof. We proceed by first constructing a candidate mapping α 0 from L 1 (H 1 ) toL 2 (H 2 ) and then show that it is an extension of α with the desired properties.Step 1: Construction of the mapping. Let X ∈ L 1 (H 1 ) with localising sequenceτ n . Theng τn (α(f τn+1 (X))) = α((f τn ◦ f τn+1 )(X)) = α(f τn (X)).The Pasting Lemma 2.5.10 yields the existence of some process Y with the propertythat g τn (Y ) = α(f τn (X)). We want to define α 0 (X) as this process Y . For this to bemeaningful, we need to check that Y does not depend on the localising sequence used.Therefore, let τ ∗ n be another localising sequence for X and let Y ∗ be the element suchthat g τ ∗ n(Y ∗ ) = α(f τ ∗ n(X)). We want to show that Y = Y ∗ . Since τ n and τ ∗ n both arelocalising sequences for X, by Lemma 2.5.8, so is τ ∗ n ∧ τ n , andg τ ∗n ∧τ n(Y ) = (g τ ∗n◦ g τn )(Y )= g τ ∗n(α(f τn (X)))= α((f τ ∗n◦ f τn )(X))= α((f τn ◦ f τ ∗n)(X))= g τn (α(f τ ∗n(X)))= (g τn ◦ g τ ∗n)(Y ∗ )= g τ ∗n ∧τ n(Y ∗ )Since τ ∗ n ∧ τ n is determining, Y = Y ∗ . Therefore, we can define α 0 : L 1 (H 1 ) → L 2 (H 2 )by putting α 0 (X) = Y . We need to show that α 0 is an extension of α and has thelocalisation property mentioned in the theorem.Step 2: α 0 is an extension of α. To show that α 0 is an extension, let an elementX ∈ H 1 be given, and let τ n be a determining sequence. Then τ n is localising for X,and therefore g τn (α 0 (X)) = α(f τn (X)) = g τn (α(X)). Thus α 0 (X) = α(X).Step 3: α 0 has the localisation property. We show g σ (α 0 (X)) = α 0 (f σ (X)) forany σ. Still letting τ n be localising for X, we know by Lemma 2.5.7 that τ n is also

2.5 Localisation 35localising for any f σ (X), and thereforeg τn (g σ (α 0 (X))) = g σ (g τn (α 0 (X)))= g σ (α(f τn (X)))= α((f σ ◦ f τn )(X))= α((f τn ◦ f σ )(X))= g τn (α 0 (f σ (X))).Since τ n is determining, we conclude g σ (α 0 (X)) = α 0 (f σ (X)).Step 4: Uniqueness. Finally, assume that α 1 and α 2 are two extensions of αsatisfying the criterion in the theorem. We want to show that α 1 = α 2 . Let anelement X ∈ L 1 (H 1 ) be given and let τ n be a localising sequence. We obtaing τn (α 1 (X)) = α 1 (f τn (X))= α(f τn (X))= α 2 (f τn (X))= g τn (α 2 (X)).Since τ n is determining, α 1 (X) = α 2 (X), as desired.Corollary 2.5.12. Let H 1 and H 2 be linear spaces. If the mapping α : H 1 → H 2 inTheorem 2.5.11 is linear and each element of the localisation concepts L 1 and L 2 arelinear, then the extension α : L 1 (H 1 ) → L 2 (H 2 ) is linear as well.Proof. By Lemma 2.5.9, L 1 (H 1 ) and L 2 (H 2 ) are linear spaces, so the conclusion ofthe corollary is well-defined. The extension satisfies g τ (α(X)) = α(f τ (x)). LetX, Y ∈ L 1 (H 1 ) and let λ, µ ∈ R. Let τ n be a localising sequence for X and Y ,then f τn (X), f τn (Y ) ∈ H 1 . Therefore, by the linearity of α on H 1 ,g τn (α(λX + µY )) = α(f τn (λX + µY ))= α(λf τn (X) + µf τn (Y ))= λα(f τn (X)) + µα(f τn (Y ))= λg τn (α(X)) + µg τn (α(Y ))= g τn (λα(X) + µα(Y )).Since τ n is determining, we may conclude that α is linear.

36 PreliminariesLemma 2.5.13. Assume that H is linear and that f τ∨σ + f τ∧σ = f τ + f σ . If H isL-stable, then L(L(H)) = L(H).Proof. Since H is L-stable, L(H) is L-stable as well, and therefore L(H) ⊆ L(L(H)).We need to show the other inclusion. Before doing so, we make an observation. Let τand σ be stopping times, and assume that f τ (X), f σ (X) ∈ H. Since H is stable andlinear, we find f τ∨σ (X) = f τ (X) + f σ (X) − f τ∧σ (X) ∈ H.Now let X ∈ L(L(H)) and let τ n be a localising sequence such that f τn (X) ∈ L(H).For any n, let σm n be a localising sequence such that f σ n m(f τn (X) ∈ H. Since σm n tendsto infinity almost surely, lim m P (σm n ≤ n) = 0 for all n ≥ 1. For any n ≥ 1, pickm(n) such that P (σm(n) n ≤ n) ≤ 12. Obviously, then ∑ ∞n n=1 P (σn m(n)≤ n) < ∞, sothe Borel-Cantelli Lemma yields σm(n)n a.s.a.s.−→ ∞. Since also τ n −→ ∞, we may concludeσm(n) n ∧ τ a.s.n −→ ∞. Now putρ n = maxk≤n σk m(k) ∧ τ k.Then ρ is a sequence of stopping times tending almost surely to infinity, and by ourearlier observation, f ρn (X) ∈ H. Therefore X ∈ L(H).The criterion in Lemma 2.5.13 is satisfied both for stopping and zero-stopping. Weare now done with general results on localisation. We proceed to give a few resultsregarding more specific matters of localisation.Definition 2.5.14. Let H be a class of stochastic proceses. We denote the continuouselements of H by cH.Lemma 2.5.15. It always holds that cL(H) = L(cH).Proof. Let X ∈ cL(H), and let τ n be a localizing sequence for X. Since we haveX τ nt = X τn∧t and X is continuous, clearly X τn is continuous as well and thereforeX ∈ L(cH). Contrarily, assume that X ∈ L(cH), and let τ n be a localizing sequencefor X. Since X τ nis continuous, X is continuous on [0, τ n ]. Because τ n tends to infinity,it follows that X is continuous and therefore X ∈ cL(H).Lemma 2.5.16. Let M be a local martingale. Then M is locally a uniformly integrablemartingale.

38 PreliminariesProof. Let τ n be a localising sequence for M and let τ be any bounded stoppingtime. Then, τ n ∧ τ is a bounded stopping time for each n, and M τn ∧τ is bounded inL p , therefore uniformly integrable. Since M is continuous, M τn ∧τ converges to M τpointwise. In particular, M τn ∧τ converges in probability to M τ . By Lemma B.5.3,M τn∧τ converges in L 1 to M τ . Therefore, EM τ = lim EM τn∧τ = 0. By Lemma2.4.13, M is a martingale.

Chapter 3The Itô IntegralIn this chapter, we will develop the theory of integration with respect to a simpleclass of stochastic processes, which nevertheless will be rich enough to cover all ofour purposes. Our goal will be to create a relatively self-contained and thoroughpresentation of the theory. We will not merely be concerned with results for use inthe later development of the Malliavin calculus and the applications in finance, butwill seek to convey a general understanding of the properties of the integral. Ourconstruction will mainly follow the methods of Steele (2000), Karatzas & Shreve(1988), Øksendal (2005) and Rogers & Williams (2000b).We will work on a filtered probability space (Ω, F, F t , P ) with a F t brownian motionW satisfying the usual conditions. Such a probability space exists according to ourresults from Section 2.1. Our first task will be to define a meaningful stochastic integralwith respect to W . The construction will be done in several steps, first defining theintegral for a simple class of integrands, and then extending the space of integrands inseveral steps.After establishing the basic theory of integration with respect to one-dimensional Brownianmotion, we will consider a n-dimensional Brownian motion and extend the integralto integrators of the form M t = ∑ n∫ tk=1 0 Y s k dWs k . The development of the integralfor this type of processes will mainly be done by reusing parts of the theory for theone-dimensional brownian case. We will later see that in a special case, this class ofintegrands actually is equal to the class of local martingales. After defining the integral

40 The Itô Integralfor these integrators, we extend the theory to integrators which also have a componentof continuous paths with finite variation. The stochastic integral of a process Y withrespect to a process X will be denoted I X (Y ) or Y · X. We also use the notation(Y · X) t = ∫ t0 Y s dX s .After the construction of the integral, we will prove some of the fundamental theoremsof the theory: the Itô formula, the Martingale Representation Theorem and Girsanov’sTheorem.Before beginning work on the stochastic integral with respect to Brownian motion,we develop some general results on the space of elementary processes that will provevery useful. This allows us a separation of concerns: We will be able to see whichelements of the theory essentially only concern the nature of the space of elementaryprocesses and which elements concern the Brownian motion specifically. We will alsoavoid letting the development of the integral be disturbed by the somewhat tediousresults for the space of elementary processes.The plan for the chapter is thus as follows.1. Develop some basic results for elementary processes.2. Construct the integral with respect to Brownian motion.3. Construct the integral with respect to integrators of type M t = ∑ n∫ tk=1 0 Y s k dWs k .4. Construct the integral with respect to integrators with finite variation.5. Establish the basic results of the theory.3.1 The space of elementary processesThe purpose of this section is twofold: First, we will show how to define the integralrigorously on the set bE of elementary processes. Second, we will prove a density resultfor bE and use this result to demonstrate how to extend a class of mappings from bEto certain L 2 spaces.Definition 3.1.1. Let Z be a stochastic variable and let σ and τ be stopping times. Wedefine the process Z(σ, τ] by Z(σ, τ] t = Z1 (σ,τ] (t). We define the class of elementary

3.1 The space of elementary processes 41processes bE by{ n }∑bE = Z k (σ k , τ k ]∣ σ k ≤ τ k are bounded stopping times and Z k ∈ bF σk .k=1Clearly, bE is a linear space. Let M be any stochastic process, and let H ∈ bE be givenwith H = ∑ nk=1 Z k(σ k , τ k ]. Intuitively, we would like to define the integral I M (H) t ofH with respect to M over [0, t] by the Riemann-sum-like expressionI M (H) t =n∑Z k (M τk ∧t − M σk ∧t).k=1In order for this definition to work, however, we must make sure that this definitiondoes not depend on the representation of H. It is clear that if H = ∑ nk=1 Z k(σ k , τ k ], wealso find H = ∑ n−1k=1 Z k(σ k , τ k ] + 1 2 Z k(σ k , τ k ] + 1 2 Z k(σ k , τ k ], or, with bounded stoppingtimes ρ k such that σ k ≤ ρ k ≤ τ k , H = ∑ n−1k=1 Z k(σ k , ρ k ] + Z k (ρ k , τ k ]. In other words,H can be represented on the form ∑ nk=1 Z k(σ k , τ k ] in many equivalent ways. Wewill now check that the definition of the stochastic integral does not depend on therepresentation.Lemma 3.1.2. Let H, K ∈ bE with H = ∑ nk=1 Z k(σ k , τ k ] and K = ∑ mk=1 Z′ k (σ′ k , τ k ′ ].There exists representations of H and K based on the same stopping times. Oneparticular such representation can be obtained by puttingH =2n+2m−1∑i=1Y i (U i , U i+1 ] and K =2n+2m−1∑i=1Y ′i (U i , U i+1 ],where U i is the i’th ordered value amongst the 2n + 2m stopping times in the family{σ 1 , . . . , σ n , τ 1 , . . . , τ n , σ ′ 1, . . . , σ ′ m, τ ′ 1, . . . , τ ′ m} andY i =n∑k=1Z k 1 (σk ≤U i

42 The Itô Integralsame lemma that Z k 1 (σk ≤U i

3.1 The space of elementary processes 43representations yield the same result. Assume that we have two different representationsof H, H = ∑ nk=1 Z k(σ k , τ k ] = ∑ mk=1 Z′ k (σ′ k , τ k ′ ]. We want to show thatn∑m∑Z k (M τk ∧t − M σk ∧t) = Z k(M ′ τ ′k ∧t − M σ ′k∧t).k=1By Lemma 3.1.2, we have H = ∑ 2n+2m−1i=1Y i (U i , U i+1 ] = ∑ 2n+2m−1i=1Y ′i (U i, U i+1 ],where U i is the i’the ordered value amongst the 2n + 2m stopping times in the family{σ 1 , . . . , σ n , τ 1 , . . . , τ n , σ ′ 1, . . . , σ ′ m, τ ′ 1, . . . , τ ′ m} andY i =n∑k=1We will show the two equalitiesk=1Z k 1 (σk ≤U i

44 The Itô IntegralTheorem 3.1.3 ensures the existence of the stochastic integral I M on bE, taking thevalues we would expect no matter what the representation of the bE-element. In thefollowing, we will interchangeably use the notation I M (H) and H · M for the integralof H with respect to M over [0, t]. We will also write I M (H) t = ∫ t0 H s dM s . Note thatusing the notation from Chapter 2 for stopped processes, we can write the processI M (H) in compact form, with H = ∑ nk=1 Z k(σ k , τ k ], byI M (H) =n∑Z k (M τ k− M σ k).k=1We also want to import meaning to the symbol ∫ tH(u) dM(u). The following lemmassshows what the proper interpretation is.Lemma 3.1.4. Let 0 ≤ x ≤ y and 0 ≤ t ≤ s. It holds that(x, y] ∩ (s, t] = (x ∧ t ∨ s, y ∧ t ∨ s],understanding that x ∧ t ∨ s = (x ∧ t) ∨ s and y ∧ t ∨ s = (y ∧ t) ∨ s.Proof. We consider the three possible cases separately.The case x ≤ y ≤ s ≤ t. We find(x, y] ∩ (s, y] = ∅(x ∧ t ∨ s, y ∧ t ∨ s] = (x ∨ s, y ∨ s] = (s, s] = ∅,so the proposition holds in this case.The case x ≤ s ≤ y ≤ t. In this case, we see that(x, y] ∩ (s, y] = (s, y](x ∧ t ∨ s, y ∧ t ∨ s] = (x ∨ s, y ∨ s] = (s, y],which verifies the statement in this case also.The case s ≤ t ≤ x ≤ y. In the final case, we obtain(x, y] ∩ (s, y] = ∅(x ∧ t ∨ s, y ∧ t ∨ s] = (t ∨ s, t ∨ s] = (t, t] = ∅,and this concludes the proof of the lemma.

3.1 The space of elementary processes 45Lemma 3.1.5. Let H ∈ bE with H = ∑ nk=1 Z k(σ k , τ k ]. It holds thatn∑Z k (M τk ∧t∨s − M σk ∧t∨s) = I M (H) t − I M (H) s .k=1Proof. It will suffice to consider the term Z k (M τk ∧t∨s − M σk ∧t∨s) for each of the threepossible configurations of the intervals (σ k , τ k ] and (s, t], and prove it equal to thecorresponding term of I M (H) t − I M (H) s .The case s ≤ t ≤ σ k ≤ τ k . We getZ k (M τk ∧t − M σk ∧t) − Z k (M τk ∧s − M σk ∧s)= Z k (M t − M t ) − Z k (M s − M s )= Z k (M τk ∧t∨s − M σk ∧t∨s).showing the statement in this case.The case s ≤ σ k ≤ t ≤ τ k . Here, we findZ k (M τk ∧t − M σk ∧t) − Z k (M τk ∧s − M σk ∧s)= Z k (M t − M σk ) − Z k (M s − M s )= Z k (M τk ∧t∨s − M σk ∧t∨s).The case s ≤ t ≤ σ k ≤ τ k . In the final case, we obtainZ k (M τk ∧t − M σk ∧t) − Z k (M τk ∧s − M σk ∧s)= Z k (M τk − M σk ) − Z k (M τk − M σk )= Z k (M τk ∧t∨s − M σk ∧t∨s),as desired.Conclusion. All in all, we conclude thatn∑Z k (M τk ∧t∨s − M σk ∧t∨s)=k=1n∑Z k (M τk ∧t − M σk ∧t) − Z k (M τk ∧s − M σk ∧s)k=1= I M (H) t − I M (H) s ,and the lemma is proved.

46 The Itô IntegralLemma 3.1.5 shows that the natural definition of the integral ∫ tH(u) dM(u) coincidesswith I M (H) t −I M (H) s , which is of course not particularly surprising. In the following,we will by ∫ ts H(u) dM(u) denote the variable I M (H) t − I M (H) s .Lemma 3.1.6. The integral mapping I M on bE is linear.Proof. Let H and K in bE be given with representationsH =and let λ, µ ∈ R. Thenn∑m∑Z k (σ k , τ k ] and K = Y i (V i , W i ]k=1i=1I M (λH + µK)( tn)∑m∑= I M λZ k (σ k , τ k ] + µY i (V i , W i ]=k=1i=1n∑m∑λZ k (M τ k(t) − M σ k(t)) + µY i (M Wi (t) − M Vi (t))k=1i=1= λI M (H) t + µI M (K) t ,which was to be shown.We have now shown how to define the integral on bE for any process M in a sensibleway, and we have checked linearity of the integral. In order to extend the integral tolarger spaces of integrands, we will need to require more structure on M. We will seehow to do this in the following sections.Next, we will show an extension result which will be crucial to the later development ofthe integral. First, we show that there is a large class of L 2 spaces on [0, ∞) × Ω whichcontain bE as a dense subspace. Recall that Σ π denotes the progressive σ-algebra on[0, ∞) × Ω.Theorem 3.1.7. Let µ be a measure on Σ π which is absolutely continuous with respectto λ ⊗ P , such that µ([0, T ] × Ω) < ∞ for all T > 0. Then bE is a dense subspace ofL 2 (µ).Proof. By ‖ · ‖ µ , we denote the standard seminorm of L 2 (µ). We first show that bE isa subspace of L 2 (µ). Let H ∈ bE be given with H = ∑ nk=1 Z k(σ k , τ k ]. Using Lemma

3.1 The space of elementary processes 472.3.4, we see that H is progressive. We find‖H‖ 2 µ =≤=n∑∫Z k (ω)1 (σk (ω),τ k (ω)](t) dµ(t, ω)k=1n∑∫‖Z k ‖ ∞ 1 [0,‖σk ‖ ∞ +‖τ k ‖ ∞ ]×Ω dµ(t, ω)k=1n∑‖Z k ‖ ∞ µ([0, ‖σ k ‖ ∞ + ‖τ k ‖ ∞ ] × Ω) < ∞,k=1as desired. The proof of the density of bE proceeds in four steps, each consisting ofshowing that a particular subset of L 2 (µ) is dense in L 2 (µ). The four subsets are:1. Processes which are zero from a point onwards.2. Processes which are zero from a point onwards and bounded.3. Processes which are zero from a point onwards, bounded and continuous.4. The set bE.Step 1: Approximation by processes which are zero from a point onwards.Let X ∈ L 2 (µ) and define X n = X1 [0,T ]×Ω . Since [0, T ] × Ω ∈ Σ π , X n is progressive.Clearly ‖X n ‖ µ ≤ ‖X‖ µ < ∞, so X n ∈ L 2 (µ). And X n is zero from T onwards. Finally,∫lim ‖X − X n ‖ 2 µ = lim 1 (T,∞)×Ω X 2 dµ = 0by dominated convergence. This means that in the following, in our work to showdensity of bE in L 2 (µ), we can assume that the process to be approximated is zerofrom a point onwards.Step 2: Approximation by bounded processes. We next show that each elementof L 2 (µ), zero from a point onwards, can be approximated by bounded processes zerofrom a point onwards. Therefore, let X ∈ L 2 (µ) be zero from T onwards and defineX n (t) = X(t)1 [−n,n] (X t ). Then X n is progressive and bounded. And we clearly have‖X n ‖ µ ≤ ‖X‖ µ < ∞, so X n ∈ L 2 (µ), and X n is zero from T onwards. Furthermore,∫lim ‖X − X n ‖ 2 µ = lim 1 (|X|>n) X 2 dµ = 0nby dominated convergence.

48 The Itô IntegralStep 3: Approximation by continuous processes. We now show that eachelement of L 2 (µ), bounded and zero from a point onwards, can be approximated bycontinuous processes of the same kind. Let X ∈ L 2 (µ) be bounded and zero from Tonwards. PutX n (t) = 1 1n∫ t(t− 1 n )+ X(s) ds,where x + = max{0, x}. Clearly, X n is continuous and ‖X n ‖ ∞ ≤ ‖X‖ ∞ < ∞, so X nis bounded. For t ≥ T + 1 n , X n(t) = 0. We will show that the sequence X n is in L 2 (µ)and approximates X.Since X is progressive, X |[0,t]×Ω is B[0, t] ⊗ F t measurable. Therefore, X n (t) is F tmeasurable. Since X n is also continuous, X n is progressive by Lemma 2.2.4. Since X nis bounded and zero from a certain point onwards, we conclude X n ∈ L 2 (µ).It remains to show convergence in ‖ · ‖ µ of X n to X. By Theorem A.2.4, X n (ω)converges Lebesgue almost surely to X(ω) for all ω ∈ Ω. This means that we haveconvergence λ⊗P everywhere. Absolute continuity yields that X n converges µ-almostsurely to X. Since X n and X have common support [0, T + 1] × Ω of finite measureand are bounded by the common constant ‖X‖ ∞ , by dominated convergence we obtainlim ‖X − X n ‖ µ = 0.Step 4: Approximation by elementary processes. Finally, we show that for eachX ∈ L 2 (µ) which is continuous, bounded and zero from some point T and onwards,we can approximate X with elements of bE.To do this, we defineX n (t) =∞∑k=0( ) kX2 n 1 [k2 n , k+1)(t).2 nSince X n is zero from T and onwards, the sum defining X n is zero from the termnumber [2 n T ] + 2 and onwards. We therefore conclude X n ∈ bE. Since k2= [2n t]n 2 nwhen k2≤ t < k+1n 2, we can also write X n n (t) = X( [2n t]2). In particular, X n n is zerofrom T + 1 and onwards. Since lim [2n t]2= t for all t ≥ 0, it follows from the continuitynof X that X n converges pointwise to X. As in the previous step, since X n and Xhas common support of finite measure and are bounded by the common constant‖X‖ ∞ , dominated convergence yields lim ‖X − X n ‖ µ = 0. This shows the claim of thetheorem.Note that in the proof of Theorem 3.1.7, we actually showed a stronger result than

3.1 The space of elementary processes 49just that bE is dense in L 2 (µ). Our final stage of approximations were with processesin the set{ n }∑bS = Z k (s k , s k ]∣ s k ≤ t k are deterministic times and Z k ∈ bF σkk=1so we have in fact demonstrated that bS is dense in L 2 (µ), even though we will notneed this stronger fact.In our coming definition of the integral, we will have at our disposal an integral mappingI : bE → cM 2 0 which is linear and isometric, when bE is considered as a subspaceof a suitable space L 2 space on Σ π . The following theorem shows that such mappingsalways can be extended from bE to the L 2 space.Theorem 3.1.8. Let µ be a measure on Σ π , absolutely continuous with respect toλ ⊗ P and such that µ([0, T ] × Ω) < ∞ for all T > 0. Let I : bE → cM 2 0 be linear andisometric, considering bE as a subspace of L 2 (µ). There exists a linear and isometricextension of I to L 2 (µ). This mapping is unique up to indistinguishability.Comment 3.1.9 The linearity stated in the theorem is to be understood as linearityup to indistinguishability. This kind of “up to indistinguishability” qualifier will beimplicit in much of the following.◦Proof. First, we construct the mapping. Let X ∈ L 2 (µ). By Theorem 3.1.7, bEis dense in L 2 (B), so there exists a sequence H n in bE converging to X in L 2 (µ).In particular, H n is a cauchy sequence in L 2 (µ). By the isometry property of I onbE, I(H n ) is then a cauchy sequence in cM 2 0. Since cM 2 0 is complete by Lemma2.4.9, there exists Y ∈ cM 2 0 such that I(H n ) converges to Y , and Y is unique up toindistinguishability. We now have a candidate for I(X). However, we need to makesure that this candidate does not depend on the sequence in bE approximating X.Assume therefore that K n is another such sequence, and let Z be a limit of I(K n ).Then, we have‖Y − Z‖ cM 20≤ lim sup ‖Y − I(H n )‖ cM 20+ ‖I(H n ) − I(K n )‖ cM 20+ ‖I(K n ) − Z‖ cM 20= lim sup ‖I n (H n − K n )‖ cM 20= lim sup ‖H n − K n ‖ cM 20,and since lim sup ‖H n − K n ‖ µ ≤ lim sup ‖H n − X‖ µ + ‖X − K n ‖ µ = 0, we concludethat for each X ∈ L 2 (µ), there is an element Y of cM 2 0, unique up to indistinguisha-

50 The Itô Integralbility, such that for any sequence (H n ) in bE converging to X in L 2 (µ), it holds thatlim I(H n ) = Y . We define I(X) as some element in the equivalence class of Y .Having constructed the mapping I, we need to check the properties of the mappinggiven in the theorem. To prove the isometry property, let X ∈ L 2 (µ) and let (H n ) ⊆ bEconverge to X in L 2 (µ). Then I(X) = lim I(H n ) in cM 2 0, and by the isometry propertyon bE,‖I(X)‖ cM 20= lim ‖I(H n )‖ cM 20= lim ‖H n ‖ µ = lim ‖X‖ µ .Next, we show linearity. We wish to prove that for X, Y ∈ L 2 (B) and λ, µ ∈ R, it holdsthat I(λX +µY ) = λI(X)+µI(Y ) up to indistinguishability. We know that the claimholds for elements of bE. Letting (H n ) and (K n ) be sequences in bE approximatingX and Y , respectively, we then findI(λX + µY ) = I(λ lim H n + µ lim K n )= lim I(λH n + µK n )= lim λI(H n ) + µI(K n )= λI(X) + µI(Y ),up to indistinguishability, where the limits are in L 2 (µ) and cM 2 0.This proves the statements about I. It remains to prove uniqueness of I. Assumetherefore that we have two mappings, I and J, with the properties stated in thetheorem. Obviously, I(H) = J(H) for H ∈ bE. Let X ∈ L 2 (µ) and let (H n ) be asequence in bE approximating H. Then,‖I(X) − J(X)‖ cM 20= ‖ lim I(H n ) − lim J(H n )‖ cM 20= lim ‖I(H n ) − J(H n )‖ cM 20= 0,showing that I(X) and J(X) are indistinguishable.Comment 3.1.10 The proof of the existence of the extension in Theorem 3.1.8 wasmade without reference to standard theorems. Another, less direct, method of proof,would be to define the integral modulo equivalence classes making the pseudometricspaces involved into metric spaces. In this case, results for continuous extensionsfrom the theory of metric spaces can be applied directly to obtain the extension, seeTheorem 8.16 of Carothers (2000).◦We are now ready to begin the development of the stochastic integral proper.

3.2 Integration with respect to Brownian motion 513.2 Integration with respect to Brownian motionIn this section, we define the stochastic integral with respect to Brownian motion. Theconstruction takes place in three phases. Our results from Section 2.5 and Section 3.1have paved much of the way for our work, so we will be able to concentrate on theimportant parts of the process, not having to deal with technical details. The threephases are:1. Definition of the integral on bE by a Riemann-sum definition.2. Definition of the integral on the space L 2 (W ) by a density argument.3. Definition of the integral on L 2 (W ) by a localisation argument.Let W be a one-dimensional F t Brownian motion. We begin by defining and investigatingthe stochastic integral on bE. From Theorem 3.1.3 and Lemma 3.1.6, weknow that the mapping I W from bE to the space of stochastic processes given by, forH = ∑ ∞k=1 Z k(σ k , τ k ],n∑I W (H) = Z k (W τ k− W σ k)k=1is well-defined and linear. We will now show that I W maps into the space cM 2 0 ofcontinuous square-integrable martingales zero at zero.Lemma 3.2.1. W t and W 2 t − t are F t martingales.Comment 3.2.2 It is well known that if W t and W (t) 2 − t are martingales withrespect to the filtration generated by the Brownian motion W itself. The importantstatement of the lemma is that in our case, where we have an F t Brownian motion,the martingales can be taken with respect to F t .◦Proof. We first show that W is an F t martingale. Let s ≤ t be given, it then holdsthat W t − W s = W s+t−s − W s is independent of F s and normally distributed. Inparticular, E(W t − W s |F s ) = E(W t − W s ) = 0, showing the martingale property.To show that W 2 t − t is a F t martingale, we note that for s ≤ t, we haveW 2 t − W 2 s = (W t − W s ) 2 + 2(W t − W s )W s ,

52 The Itô Integraland therefore, since W t − W s is independent of F s ,E(W 2 t − W 2 s |F s ) = E((W t − W s ) 2 |F s ) + 2W s E(W t − W s |F s )showing E(W 2 t − t|F s ) = W 2 s − s.= t − s,Theorem 3.2.3. For any H ∈ bE, I W (H) ∈ cM 2 0 and EI W (H) 2 ∞ = E ∫ ∞0H 2 t dt.Proof. The process I W (H) is clearly continuous, since W is continuous.Step 1: The martingale property. We use Lemma 2.4.13. Let H ∈ bE withH = ∑ nk=1 Z k[σ k , τ k ), we then have lim t→∞ I W (H) t = ∑ nk=1 Z k(W σk − W τk ), soI W (H) has a well-defined almost sure limit. Now let τ be any stopping time. Since τ kand σ k are bounded stopping times, optional sampling and Lemma 2.4.11 yieldsE(H · W ) τ = E=n∑k=1Z k (W τ kτ− W σ kτ )n∑E(Z k E(Wτ τ k− Wσ τ k|F σk ))k=1= 0.By Lemma 2.4.13, then, H · W is a martingale.Step 2: Square-integrability. To show that I W (H) is bounded in L 2 , it willsuffice to show the moment equality for I W (H) ∞ , since I W (H) 2 is a submartingaleand its second moments are therefore increasing. As before, let H ∈ bE withH = ∑ nk=1 Z k[σ k , τ k ). By Lemma 3.1.2, we can assume that the stopping times areordered such that τ k ≤ σ k+1 for all k < n.We findEI W (H) 2 ∞(∑ n= E Z k (W τk − W σk )= E=k=1n∑k=1 i=1) 2n∑Z k Z i (W τk − W σk )(W τi − W σi )n∑EZk(W 2 τk − W σk ) 2 +k=1k≠in∑EZ k Z i (W τk − W σk )(W τi − W σi ).

3.2 Integration with respect to Brownian motion 53Here we have, applying optional sampling for bounded stopping times and using thatW 2 t − t by Lemma 3.2.1 is a F t -martingale,EZ 2 k(W τk − W σk ) 2= EZ 2 kE((W τk − W σk ) 2 |F σk )= EZ 2 kE(W 2 τ k− τ k + W 2 σ k+ σ k − 2W τk W σk |F σk ) + EZ 2 kE(τ k − σ k |F σk )= EZ 2 kE(W 2 τ k− τ k − (W 2 σ k− σ k )|F σk ) + EZ 2 k(τ k − σ k )= EZ 2 k(τ k − σ k ).Furthermore, for i < k we find that, again by optional sampling, using the ordering ofthe stopping times,Finally, we concludeas desired.EZ k Z i (W τk − W σk )(W τi − W σi )= EZ i (W τi − W σi )Z k E((W τk − W σk )|F σk )= 0.EI W (H) 2 ∞ =n∑EZk(τ 2 k − σ k ) = Ek=1∫ ∞0H 2 u du,We have now proven that I W maps bE into cM 2 0. If we can embed bE into a L 2 -spacesuch that I W becomes an isometry, we can use Theorem 3.1.8 to extend I W to thisspace.Definition 3.2.4. By L 2 (W ) we denote the set of adapted and measurable processesX such that E ∫ ∞0X 2 t dt < ∞. L 2 (W ) is then equal to the L 2 space of square integrablefunctions on [0, ∞) × Ω with the progressive σ-algebra Σ π under the measure λ ⊗ P .We denote the L 2 seminorm on L 2 (W ) by ‖ · ‖ W .Theorem 3.2.5. There exists a linear isometric mapping I W : L 2 (W ) → cM 2 0 suchthat for H ∈ bE, with H = ∑ nk=1 Z k[σ k , τ k ), I W (H) = ∑ nk=1 Z k(W τ k− W σ k). Thismapping is unique up to indistinguishability.Proof. Considering bE as a subspace of L 2 (W ), we have by Theorem 3.2.3 for H ∈ bEthat ‖I W (H)‖ 2 = ‖IcM 2 W (H) ∞ ‖ 2 2 = E ∫ ∞H(t) 2 dt = ‖H‖ 200 W , so I W : bE → cM 2 0 isisometric. Since it is also linear, the conditions of Theorem 3.1.8 are satisfied, and theconclusion follows.

54 The Itô IntegralComment 3.2.6 The isometry property of I W is known as Itô’s isometry.◦We still need to make a final extension of the stochastic integral with respect toBrownian motion. This extension is done by the technique of localisation. It is duringthis step that the results of Section 2.5 will prove worth their while.The basic idea of the localisation is that if a process is in L 2 (W ) when we cut it offand put it to zero from a certain point onwards, then we can define the integral up tothis time. If the cutoff times tend to infinity, we can paste the integrals together andobtain a reasonable stochastic integral.The cutoff procedure will, in order to obtain an appropriate level of generality, bedone at a stopping time. We will need two kinds of localization, described in Section2.5, namely stopping and zero-stopping. We review the localisation concepts herefor completeness. We say that a process M is locally, or L-locally, in cM 2 0 if thereis a sequence of stopping times τ n increasing to infinity such that M τ n, defined byM τ nt = M τn ∧t, is in cM 2 0. We say that a process X is zero-locally, or L 0 -locally,in L 2 (W ) if there is a sequence of stopping times τ n increasing to infinity such thatX[0, τ n ], defined by X[0, τ n ] t = X t 1 [0,τ] (t), is in L 2 (W ).We denote the processes L-locally in cM 2 0 by cM L 0 , and we denote the processes L 0 -locally in L 2 (W ) by L 2 (W ). We call cM L 0 the space of continuous local martingales.We are going to use Theorem 2.5.11 to prove that there is a unique extension of I Wfrom L 2 (W ) → cM 2 0 to L 2 (W ) → cM L 0 such that I W (X) τ = I W (X[0, τ]) for anystopping time τ. We will therefore need to check the hypotheses of this theorem.Lemma 3.2.7. The spaces L 2 (W ) and L 2 (W ) are linear and L 0 -stable. The spacescM 2 0 and cM L 0 are linear and L-stable.Proof. By Lemma 2.5.18, L 2 (W ) is L 0 -stable, and it is clearly linear. By Lemma 2.5.9,L 2 (W ) is linear and L 0 -stable. By Lemma 2.4.11, martingales are L-stable. Then cM 2 0is also L-stable. Since cM 2 0 is also linear, Lemma 2.5.9 yields that cM L 0 is linear.Theorem 3.2.8. Let X ∈ L 2 (W ) and let τ be a stopping time. Then it holds thatI(X) τ = I(X[0, τ]).Proof. We first show that the result holds for bE, and then extend by a density argument.

3.2 Integration with respect to Brownian motion 55Step 1: The elementary case. Let H ∈ bE with H = ∑ nk=1 Z k(σ k , τ k ]. We havethatn∑n∑H[0, τ](t) = Z k 1 (σk ,τ k ](t)1 [0,τ] (t) = Z k 1 (σk ≤τ)1 (σk ∧τ,τ k ∧τ](t),k=1and by Lemma 2.3.8, Z k 1 (σk ≤τ) ∈ F σk ∧τ . Therefore, H[0, τ] ∈ bE, and we haveI(H[0, τ]) t ==k=1n∑Z k 1 (σk ≤τ)(W τk ∧τ∧t − W σk ∧τ∧t)k=1n∑Z k (W τk ∧τ∧t − W σk ∧τ∧t)k=1= I(H) τ t .Step 2: The general case. Now let X ∈ L 2 (W ) be arbitrary and let X n be asequence in bE converging to X. We trivially have ‖X n [0, τ]−X[0, τ]‖ W ≤ ‖X n −X‖ W ,so X n [0, τ] converges to X[0, τ] in L 2 (W ). Since I(X n [0, τ]) = I(X n ) τ , the isometryproperty of the integral yields‖I(X[0, τ]) − I(X) τ ‖ cM 20≤ ‖I(X[0, τ]) − I(X n [0, τ])‖ cM 20+ ‖I(X n ) τ − I(X) τ ‖ cM 20≤ ‖I(X[0, τ]) − I(X n [0, τ])‖ cM 20+ ‖I(X n ) − I(X)‖ cM 20= ‖X[0, τ] − X n [0, τ]‖ W + ‖X n − X‖ W≤ 2‖X n − X‖ W .Thus, I(X τ ) = I(X) τ up to indistinguishability, as desired.With Theorem 3.2.8 in hand, we are ready to extend the integral.Theorem 3.2.9. There exists an extension of I W : L 2 (W ) → cM 2 0 to a mappingfrom L 2 (W ) to cM L 0 . The extension is uniquely determined by the criterion that(X · W ) τ = X[0, τ] · W for any X ∈ L 2 (W ). The extension is linear.Proof. We apply Theorem 2.5.11 with the localisation concepts L 0 of zero-stoppingand L of stopping. From Lemma 3.2.7, we know that L 2 (W ) is L 0 -stable and cM 2 0is L-stable. And by Theorem 3.2.8, the localisation hypothesis of Theorem 2.5.11 issatisfied. Theorem 2.5.11 therefore immediately yields the desired extension, and byCorollary 2.5.12, the extension is linear.

56 The Itô IntegralTheorem 3.2.9 yields the “pasting” definition of the integral of processes in L 2 (W )discussed earlier. For general X ∈ L 2 (W ) with localising sequence τ n , the propertyI W (X) τ n= I W (X[0, τ n ]) relates the values of the integral for general X to the valuesof the integral on L 2 (W ).We have now defined the stochastic integral with respect to Brownian motion for asmany processes as we will need. Our next goal is to extend the integral to other integratorsthan Brownian motion. Before this, we summarize our results: Our stochasticintegral I W with respect to Brownian motion is a linear mapping from L 2 (W ) to cM L 0 .It has the following properties:1. For X ∈ L 2 (W ), X · W is a square-integrable martingale.2. For H ∈ bE with H = ∑ nk=1 Z k(σ k , τ k ], H · W = ∑ nk=1 Z k(W τ k− W σ k).3. For any X ∈ L 2 (W ) and any stopping time τ, (X · W ) τ = X[0, τ] · W .3.3 Integration with respect to integrators in M L WWe now use the results from Section 3.2 on integration with respect to Brownianmotion to define the stochastic integral for larger classes of integrators and integrands.We begin with a lemma to set the mood. We still denote by W a one-dimensional F tBrownian motion.Lemma 3.3.1. Let H ∈ bE with H = ∑ ∞k=1 Z k(σ k , τ k ] and let Y ∈ L 2 (W ). ThenHY ∈ L 2 (W ) and ∑ nk=1 Z k((Y · W ) τ k− (Y · W ) σ k) = HY · W .Proof. Since H is bounded and Σ π -measurable, it is clear that HY ∈ L 2 (W ). To provethe other statement, note that by linearity, it will suffice to consider H of the formH = Z(σ, τ].Step 1: The case Y ∈ bE. Assume Y = ∑ nk=1 X k(σ k , τ k ], we then findZ(σ, τ]Y ==n∑Z(σ, τ]X k (σ k , τ k ]k=1n∑ZX k 1 (σk ≤τ)(σ k ∧ τ ∨ σ, τ k ∧ τ ∨ σ],k=1

3.3 Integration with respect to integrators in M L W 57by Lemma 3.1.4, where X k 1 (σk ≤τ) is F σk ∧τ measurable by Lemma 2.3.8, so ZX k 1 (σk ≤τ)is F σk ∧τ∨σ measurable. This shows Z(σ, τ]Y ∈ bE, and we may then concludeZ((Y · W ) τ − (Y · W ) σ )( n)∑n∑= Z X k (W τk∧τ − W σk∧τ ) − X k (W τk∧σ − W σk∧σ )===k=1k=1n∑ZX k (W τk∧τ − W σk∧τ − W τk∧σ + W σk∧σ )k=1n∑ZX k (W τk∧τ∨σ − W σk∧τ∨σ )k=1n∑ZX k 1 (σk ≤τ) (W τk∧τ∨σ − W σk∧τ∨σ )k=1= HY · W.Step 2: The case Y ∈ L 2 (W ). Assuming Y ∈ L 2 (W ), let H n be a sequence inbE converging towards Y . Since H is bounded, lim ‖HY − HH n ‖ W = 0. From thecontinuity of the integral, H n ·W converges to Y ·W . Therefore, (H n ·W ) τ −(H n ·W ) σconverges to (Y · W ) τ − (Y · W ) σ . Because Z is bounded, we obtainZ((Y · W ) τ − (Y · W ) σ ) = limnZ((H n · W ) τ − (H n · W ) σ )where the limits are in L 2 (W ).= limnHH n · W= limnHY · W,Step 3: The case Y ∈ L 2 (W ). Let τ n be a localising sequence for Y . Then(Z(Y · W ) τ − (Y · W ) σ ) τ = Z((Y [0, τ n ] · W ) τ − (Y [0, τ n ] · W ) σ )= HY [0, τ n ] · W= (HY · W ) τ n,and letting τ n tend to infinity, we obtain the result.What Lemma 3.3.1 shows is that if we consider Y ·W as an integrator, then the integralof an element H ∈ bE based on the Riemann-sum definition of Section 3.1 satisfies

58 The Itô IntegralH · (Y · W ) = HY · W . We call this property the associativity of the integral. We willuse this as a criterion to generalize the stochastic integrals to integrators of the formY · W where Y ∈ L 2 (W ). The most obvious thing to do would be to define, givenM = Y · W with Y ∈ L 2 (W ), the integral of a process X with XY ∈ L 2 (W ) withrespect to M by X ·M = XY ·W . This is what we will do, but with a small twist. Untilnow, in this section and the preceeding, we have only considered a one-dimensionalF t Brownian motion. In our financial applications, we are going to need to work withmultidimensional Brownian motion, and we are going to need integrators that dependon several coordinates of such a multidimensional Brownian motion at once. Therefore,we will now consider an n-dimensional F t Brownian motion W = (W 1 , . . . , W n ), asdescribed in Definition 2.1.5. Our goal will be to define the stochastic integral withrespect to integrators of the formM t =n∑k=1∫ t0Y ks dW k sby associativity, as described above. We maintain the definitions of Section 3.2 byletting L 2 (W ) denote the Σ π measurable elements of Y ∈ L 2 (λ ⊗ P ) and lettingL 2 (W ) denote the processes L 0 -locally in L 2 (W ). The next lemma shows that each ofthe coordinate processes fall under the category we worked with in the last section.Lemma 3.3.2. If W = (W 1 , . . . , W n ) is an n-dimensional F t Brownian motion, thenfor each k ≤ n, W k is a one-dimensional F t Brownian motion.Proof. Let t ≥ 0. We know that s ↦→ W t+s − W t is a n-dimensional Brownian motionwhich is independent of F t . Therefore s ↦→ Wt+s k − Wtk is a one-dimensional Brownianmotion independent of F t .In the following, whenever we write Y ∈ L 2 (W ) n , we mean that Y is an n-dimensionalproces, Y = (Y 1 , . . . , Y n ), such that Y k ∈ L 2 (W ) for each k. This notation is ofcourse also extended to other spaces than L 2 (W ).Definition 3.3.3. Let cM L W denote the class of processes M such that there existsY ∈ L 2 (W ) n with M t = ∑ n∫ tk=1 0 Y s k dWs k . In this case, we also write M = Y · W .In order to define the integral with respect to processes in cM L W by associativity, weneed to check that the integrand processes Y k in the representation of elements ofare unique. This is our next order of business.cM L W

3.3 Integration with respect to integrators in M L W 59Lemma 3.3.4. For any i, j ≤ n with i ≠ j, W i W j is a martingale.Proof. Since the conditional distribution of (Wt i − Ws, i W j t − Ws j ) given F s is twodimensionalnormal with mean zero and zero covariation, we haveE(W i t W j t + W i sW j s |F s )= E((W i t − W i s)(W j t − W j s )|F s ) + E(W i t W j s |F s ) + E(W i sW j t |F s )= W j s E(W i t |F s ) + W i sE(W j t |F s )= 2W j s W i s,so E(W i t W j t |F s ) = W i sW j s , and W i W j is a martingale.Lemma 3.3.5. Let i, j ≤ n with i ≠ j. For any Y i , Y j ∈ L 2 (W ), (Y i · W i )(Y j · W j )is a uniformly integrable martingale.Proof. We use Lemma 2.4.13. Clearly, (Y i · W i )(Y j · W j ) has an almost sure limit. Itwill therefore suffice to show that for any stopping time τ,E∫ τ0Y is dW i s∫ τ0Y js dW j s = 0.Step 1: The elementary case. First consider the case where Y i , Y j ∈ bE withWe then find= E==∫ τE0( n∑n∑Y i =k=1k=1 m=1n∑n∑Zk(σ i k , τ k ] and Y j =k=1∫ τYs i dWsin∑n∑k=1 m=10Z i k((W i ) τ k τY js dW j sEZ i kZ j m((W i ) τ k τEZ i kZ j m((W i ) τ k τ− (W i ) σ kτ )) ( n∑k=1n∑Z j k (σ k, τ k ].k=1Z j k ((W j ) τ k τ− (W j ) σ kτ )− (W i ) σ kτ )((W j ) τ m τ − (W j ) σ m τ )− (W i ) σ kτ )((W j ) τmτ − (W j ) σmτ ).Now, by Lemma 3.3.4, W i W j is a martingale, and therefore (W i W j ) τ is also a martingale.Using the optional stopping theorem, we therefore find that the above is zero.We may then conclude E ∫ τ0 Y ∫s i dWsi τ0 Y s j dWs j = 0.)

60 The Itô IntegralStep 1: The general case. In the case of general Y i , Y j ∈ L 2 (W ), there existssequences (H n ) and (K n ) in bE converging to Y i and Y j , respectively. By Lemma3.1.2, we may assume that H n and K n are based on the same stopping times, andtherefore, by what we have just shown,E∫ τ0H n (s) dW i s∫ τ0K n (s) dW j s = 0.Now, since ∫ τ0 H n(s) dWs i converges to ∫ τ0 Y s i dWsi and ∫ t0 K n(s) dWs j converges to∫ τ0 Y s j dWsj in L 2 (W ) by the Itô isometry, Lemma B.2.1 yields that the product convergesin L 1 and therefore∫ τ ∫ τ∫ τ∫ τE Ys i dWsi Ys j dWs j = 0 = lim E H n (s) dWsi K n (s) dWs j = 0,00n00as desired.Lemma 3.3.6. If Y ∈ L 2 (W ) n , then E(Y · W ) 2 ∞ = ∑ nk=1 ‖Y k ‖ 2 W .Proof. By Lemma 3.3.5, the Itô isometry and optional sampling for uniformly integrablemartingales,E( n∑k=1∫ ∞0Y ks dW k s) 2===n∑n∑Ek=1 i=1∫ ∞0n∑(∫ ∞Ek=10n∑‖Y k ‖ 2 W .k=1Y ks dW k s) 2Ys k dWsk∫ ∞0Y is dW i sLemma 3.3.7. Assume M ∈ cM L W with M = ∑ nk=1 Y ks dW k s and M = ∑ nk=1 Zk s dW k s .Then Y k and Z k are equal λ ⊗ P almost surely for k ≤ n.Proof. First consider the case where Y k , Z k ∈ L 2 (W ) for k ≤ n. We then find, byLemma 3.3.6,E( n∑k=1∫ ∞0Y ks dW k s −n∑k=1∫ ∞0Z k s dW k s) 2= E=( n∑n∑k=1k=1∫ ∞0Y ks‖Y ks − Z k s ‖ 2 W .− Z k s dW k s) 2

62 The Itô Integralprocesses, that it preserves stopping in an appropriate manner, and associativity. Afterthis, we will work to understand under which conditions the integral is a squareintegrablemartingale. We will also develop some approximation results that will beuseful in our work with the integral. In the following, let M ∈ cM L W with M = Y · W ,Y ∈ L 2 (W ) n , unless anything else is implied.It is clear that cM L W and L2 (M) are linear spaces, and by I M (X) = ∑ nk=1 I W k(XY k ),it is immediate that I M is linear for any M ∈ cM L W .Lemma 3.3.10 (Riemann representation). If H ∈ bE with H = ∑ mk=1 Z k(σ k , τ k ],H ∈ L 2 (M) and H · M = ∑ mk=1 Z k(M τ k− M σ k).Proof. By Lemma 3.3.1, HY k ∈ L 2 (W k ) for all k ≤ n, so H ∈ L 2 (M), and(H · M) t ====n∑i=1n∑∫ t0i=1 k=1m∑ ∑nZ kk=1H s Y is dW i sm∑Z k ((Y i · W i ) τ k− (Y i · W i ) σ k)i=1(Y i · W i ) τ k− (Y i · W i ) σ km∑Z k (M τ k− M σ k),k=1as desired.For Y ∈ L 2 (W ) n , we write Y [0, τ] = (Y 1 [0, τ], . . . , Y n [0, τ]). From Theorem 3.2.9, itis clear that (Y · W ) τ = Y [0, τ] · W .Lemma 3.3.11 (Localisation properties). The spaces cM L W and L2 (M) are stableunder stopping and zero-stopping, respectively, and L 2 (M) ⊆ L 2 (M τ ). It holds that(X · M) τ = X[0, τ] · M = X · M τ .Proof. cM L W is L-stable since (Y ·W )τ = Y [0, τ]·W and Y [0, τ] ∈ L 2 (W ) n by Lemma3.2.7. L 2 (M) is L 0 -stable because, if X ∈ L 2 (M), X[0, τ]Y k = (XY k )[0, τ], which isin L 2 (W ) by Lemma 3.2.7. To show that L 2 (M) ⊆ L 2 (M τ ), consider X ∈ L 2 (M).Then XY ∈ L 2 (W ) n , so XY [0, τ] ∈ L 2 (W ) n . Because M τ = Y [0, τ] · W , X ∈ L 2 (M τ )

3.3 Integration with respect to integrators in M L W 63follows. We then obtain (X · M) τ = X[0, τ] · M from(X · M) τ = (XY · W ) τ= (XY )[0, τ] · W= X[0, τ]Y · W= X[0, τ] · M.And the equality (X · M) τ = X · M τ follows since(X · M) τ = (XY · W ) τ= XY [0, τ] · W= X · M τ .Lemma 3.3.12 (Associativity). If Z ∈ L 2 (X · M), then ZX ∈ L 2 (M) and it holdsthat Z · (X · M) = ZX · M.Proof. We know that X · M = XY · W , so Z ∈ L 2 (X · M) means ZXY ∈ L 2 (W ) n ,which is equivalent to ZX ∈ L 2 (M). We then obtainas desired.Z · (X · M) = Z · (XY · W )= ZXY · W= ZX · MThe linearity, the localisation properties of Lemma 3.3.11 and the associativity ofLemma 3.3.12 are the basic rules for manipulating the integral. Our next goal isto identify classes of integrands such that the integral becomes a square-integrablemartingale. For M ∈ cM L W with M = Y · W , Y ∈ L2 (W ) n , let L 2 (M) be the spaceof Σ π measurable processes X such that XY ∈ L 2 (W ) n .Lemma 3.3.13. L 2 (M) is L 0 -stable, and L 0 (L 2 (M)) = L 2 (M).Proof. We first show that L 2 (M) is L 0 -stable. Assume that X ∈ L 2 (M), such thatXY ∈ L 2 (W ) n , and let τ be a stopping time. Then X[0, τ]Y = (XY )[0, τ] ∈ L 2 (W ),so X[0, τ] ∈ L 2 (M). Therefore, L 2 (M) is L 0 -stable.

64 The Itô IntegralTo prove the other statement of the lemma, assume X ∈ L 0 (L 2 (M)) and let τ n bea localising sequence, X[0, τ n ] ∈ L 2 (M). Since X[0, τ n ] is Σ π measurable, so is X.By definition, X[0, τ n ]Y ∈ L 2 (W ) n , so (XY )[0, τ n ] ∈ L 2 (W ) n , and it follows thatXY ∈ L 2 (W ) n , showing X ∈ L 2 (M) and therefore L 0 (L 2 (M)) ⊆ L 2 (M). Conversely,assume X ∈ L 2 (M). Then XY ∈ L 2 (W ) n , so there is a localising sequence τ n yielding(XY )[0, τ n ] ∈ L 2 (W ) n . Then X[0, τ n ]Y ∈ L 2 (W ) n and X[0, τ n ] ∈ L 2 (M), showingX ∈ L 0 (L 2 (M)).Lemma 3.3.14. L 2 (M) is equal to L 2 ([0, ∞) × Ω, Σ π , µ M ), where µ M is the measurehaving density ∑ nk=1 (Y k ) 2 with respect to λ ⊗ P . We endow L 2 (M) with the L 2 -normcorresponding to µ M and denote this norm by ‖ · ‖ M .Proof. Let X ∈ L 2 (M). Then X is Σ π measurable and XY ∈ L 2 (W ) n , so X 2 (Y k ) 2is integrable with respect to λ ⊗ P . Thus L 2 (M) ⊆ L 2 ([0, ∞) × Ω, Σ π , µ M ). On theother hand, let X ∈ L 2 ([0, ∞) × Ω, Σ π , µ M ). Then X is obviously Σ π measurable and∫‖XY k ‖ 2 W =∫X 2 (Y k ) 2 d(λ ⊗ P ) ≤X 2n ∑k=1∫(Y k ) 2 d(λ ⊗ P ) =X 2 dµ M < ∞,so X ∈ L 2 (M) and therefore L 2 ([0, ∞) × Ω, Σ π , µ M ) ⊆ L 2 (M).Comment 3.3.15 Note that since we only require Y ∈ L 2 (W ) n and not Y ∈ L 2 (W ) n ,the measure µ may easily turn out to be unbounded.◦Lemma 3.3.16. Let M ∈ cM L W and X ∈ L2 (M). X · M is a square-integrablemartingale if and only if X ∈ L 2 (M).Proof. If X ∈ L 2 (M), XY ∈ L 2 (W ) n and therefore, by linearity of cM 2 0,X · M = XY · W =n∑XY k · W k ∈ cM 2 0.k=1To prove the other implication, assume that X · M ∈ cM 2 0. Because X ∈ L 2 (M), weknow that XY ∈ L 2 (W ) n . Let τ m be a localising sequence with XY k [0, τ m ] ∈ L 2 (W )for k ≤ n. Since XY · W = X · M ∈ cM 2 0, (XY · W ) τ m∈ cM 2 0 as well. The Itô

3.3 Integration with respect to integrators in M L W 65isometry from Lemma 3.3.6 yields‖(XY k )[0, τ m ]‖ 2 W≤≤n∑‖XY k [0, τ m ]‖ 2 Wk=1n∑‖XY k ‖ 2 Wk=1= ‖XY · W ‖ 2 cM 2 0= ‖X · M‖ 2 cM 2 0,where the last expression is finite by assumption. Now, monotone convergence yields‖XY k ‖ W = lim sup ‖(XY k )[0, τ m ]‖ W ≤ ‖X · X‖ cM 20< ∞,mso XY k ∈ L 2 (W ) and thus X ∈ L 2 (M).Lemma 3.3.17. I M : L 2 (M) → cM 2 0 is isometric for any M ∈ cM L W .Proof. The conclusion is well-defined since I M (X) ∈ cM 2 0 for X ∈ L 2 (M) by Lemma3.3.16. To prove the result, we merely note for X ∈ L 2 (M) that by Lemma 3.3.6‖X · M‖ 2 cM 2 0= ‖XY · W ‖ 2 cM 2 0n∑= ‖XY k ‖ 2 W==k=1n∑∫X 2 (Y k ) 2 d(λ ⊗ P )k=1∫X 2 dµ M= ‖X‖ 2 M .Comment 3.3.18 The isometry property of I M is, as the isometry property of I Wgiven in Theorem 3.2.5, also known as Itô’s isometry.◦Lemma 3.3.16 solves the problem of determining when the stochastic integral is asquare-integrable martingale, and Lemma 3.3.17 tells us that for general integrators,we still have the isometry property that we proved in the case of Brownian motion.We have yet one more property of the integral that we wish to investigate. We are

66 The Itô Integralinterested in understanding when the integral process X · M can be approximated byintegrals of the form H n · M for H n ∈ bE. To understand this, we define cM 2 W as thesubclass of processes in cM L W of M = Y · W where Y ∈ L2 (W ) n .Lemma 3.3.19. cM 2 W is L-stable, and L(cM2 W ) = cML W .Proof. That cM 2 Wis L-stable follows since L2 (W ) is L 0 -stable. To show the otherstatement of the lemma, assume M ∈ cM L W , M = Y · W with Y ∈ L2 (W ) n . Let τ mbe a localising sequence such that Y [0, τ m ] ∈ L 2 (W ) n . Then M τm = Y [0, τ m ] · W , soM is locally in cM 2 W , showing cML W ⊆ L(cM2 W ). Conversely, assume M ∈ L(cM2 W )and let τ n be a localising sequence. Then we have M τ n∈ cM 2 W , and there existsY n ∈ L 2 (W ) such that M τ n= Y n · W . Clearly, thenY n+1 [0, τ n ] · W = (Y n+1 · W ) τn= (M τn+1 ) τn= M τn= Y n · W.In detail, this means ∑ nk=1 Y n+1[0, τ n ] · W k = ∑ nk=1 Y n k · W k . By Lemma 3.3.7,Y n+1 [0, τ n ] and Y n are then λ ⊗ P almost surely equal. Then the Pasting Lemma2.5.10 yields processes Y k such that Y k [0, τ n ] and Yn k [0, τ n ] are λ ⊗ P almost surelyequal for all n. In particular, Y ∈ L 2 (W ) n , and clearlyM τ n= Y n [0, τ n ] · W = Y [0, τ n ] · W = (Y · W ) τ n,so M = Y · W and L(cM 2 W ) ⊆ cML W .Lemma 3.3.20. If M ∈ cM 2 W , then bE is dense in L2 (M). In particular, for anyX ∈ L 2 (M), X · M can be approximated by elements H n · M with H n ∈ bE.Proof. Since M ∈ cM 2 W , M = Y · W for some Y ∈ L2 (W ) n . In particular, µ M is abounded measure, and by Theorem 3.1.7, bE is dense in L 2 (M). By Lemma 3.3.17,when X ∈ L 2 (M) and (H n ) ⊆ bE converges to X, we then obtain that H n · M alsoconverges to X · M.We are now done with our preliminary investigation of the stochastic integral forintegrators in cM L W . Our next task will be to extend the integral to integrators witha component of finite variation. Before doing so, let is review our results so far. We

3.4 Integration with respect to standard processes 67have defined the stochastic integral X · M for integrators M ∈ cM L WX ∈ L 2 (M). We have proven thatand integrands• The integral is always a continuous local martingale.• The interal is in cM 2 0 if and only if X ∈ L 2 (M).• The integral is linear in the integrand.• I M is isometric from L 2 (M) to cM 2 0.• (X · M) τ = X[0, τ] · M = X · M τ .• bE ⊆ L 2 (M), and the integral takes its natural values on bE.• If M ∈ cM 2 W , µ M is bounded and bE is dense in L 2 (M).Note that nowhere in the above properties is the Brownian motion W , which ourwhole construction rests upon, mentioned. This fact will enable us to manipulate thestochastic integral without having to keep in mind the underlying definition in termsof the Brownian motion.3.4 Integration with respect to standard processesWe now come to the final extension of the stochastic integral, where we extend theintegral to cover integrators of the form A + M, where M ∈ cM L W and A is adaptedwith continuous paths of finite variation. Since we already have defined the integralwith respect to M ∈ cM L W , we only have to define the integral with respect to A andmake sure that the integrals with respect to A and M fit properly together.Because functions of finite variation correspond to measures, as can be seen fromAppendix A.1, we can use ordinary measure theory to define the integral with respectto A in a pathwise manner. In Appendix A.1 we have listed some fundamental resultsregarding functions of finite variation. For any mapping F : [0, ∞) → R, we have thevariation function,n∑V F (t) = sup |F (t k ) − F (t k−1 )|k=1

68 The Itô Integralwhere the supremum is over all partitions of [0, t]. V F is finite for all t if and onlyif F is of finite variation. By Lemma A.1.1, if F is continuous, then so is V F . ByTheorem A.1.2, we can associate to F a measure ν F such that ν F (a, b] = F (b) − F (a)for 0 ≤ a ≤ b, and as in Definition A.1.4, we can then define L 1 (F ) = L 1 (ν F ) andconsider the integral with respect to F by putting ∫ ∞x(s) dF0 s = ∫ ∞x(s) dν0 F (s).We now transfer these ideas to stochastic processes. In order to make sure that ourstochastic integral with respect to processes of finite variation has sufficient measurabilityproperties, we will need to make the appropriate measurability assumptions onour integrands. By cFV, we denote the class of adapted stochastic processes withcontinuous paths of finite variation. If A ∈ cFV, we denote by L 1 (A) the space ofprogressively measurable processes such that X(ω) ∈ L 1 (A(ω)) for all ω. By L 1 (A),we denote the space of processes locally in L 1 (A).We begin with a result which shows that we do not need to consider integrators whichare only locally of finite variation, since this would add nothing new to the theory.Lemma 3.4.1. cFV is a linear space, stable under stopping, and L(cFV) = cFV.Proof. It is clear that cFV is a linear space. We first show that cFV is stable understopping. Let A ∈ cFV be given, and let τ be any stopping time. Clearly, A τ hascontinuous paths. By Lemma 2.3.3, A τ is progressive, therefore adapted. Fix ω. Then,for any partition (t 0 , . . . , t n ) of [0, t],n∑|A τ t k(ω) − A τ t k−1(ω)| =k=1n∑|A tk ∧τ (ω) − A tk−1 ∧τ (ω)| ≤ V A(ω) (t),k=1so taking supremum over all partitions, we conclude V A τ (ω)(t) ≤ V A(ω) , and thereforeA τ (ω) has finite variation as well, thus A τ ∈ cFV and cFV is stable under stopping.In particular, cFV ⊆ L(cFV), so to show equality it will suffice to show the otherinclusion. To this end, let A ∈ L(cFV), and let τ n be a localising sequence. Fix ω, lett > 0 and let n be so large that τ n (ω) ≥ t. Then V A(ω) (t) ≤ V A(ω) (τ n ) = V A τn (ω)(τ n ),which is finite, and we conclude that A(ω) has finite variation, showing A ∈ cFV.Next, we define the stochastic integral with respect to integrators A ∈ cFV andintegrands in L 1 (A). To show the adaptedness of the integral, we will need the conceptof a Markov kernel and the associated results, see Chapter 20 of Hansen (2004a) forthis.

3.4 Integration with respect to standard processes 69Lemma 3.4.2. If X ∈ L 1 (A), the integral ∫ t0 X s(ω) dA s (ω) is well-defined as a pathwiseintegral. We write (X · A) t (ω) = ∫ t0 X s(ω) dA s (ω).Proof. Let τ n be a localising sequence for X and let ω ∈ Ω. Let n be so large thatτ n (ω) ≥ t, then∫ t0|X s (ω)| dA s (ω) ≤∫ t0|X[0, τ n ] s (ω)| dA s (ω) ≤∫ ∞0|X[0, τ n ] s (ω)| dA s (ω),which is finite since X[0, τ n ](ω) ∈ L 1 (A(ω)). Thus, the integral ∫ t0 X s(ω) dA s (ω) iswell-defined for all ω ∈ Ω.Lemma 3.4.3. Let X ∈ L 1 (A). The stochastic integral X · A is in cFV.Proof. First assume that X ∈ L 1 (A). We need to show that X · A is adapted, continuousand of finite variation. By Lemma A.1.3, the measure corresponding to A(ω)has no atoms, and therefore X · A is continuous. By Lemma A.1.6, X · A has paths offinite variation.To show that X ·A is adapted, let t > 0 and let µ(t, ω) be the measure on [0, t] inducedby A(ω) on [0, t]. We wish to show that (µ(t, ω)) ω∈Ω is a (Ω, F t ) Markov kernel on([0, t], B[0, t]). We therefore need to show that ω ↦→ µ(t, ω)(B) is F t measurable forany B ∈ B[0, t]. The family of elements of B[0, t] for which this holds is a Dynkin class,and it will therefore suffice to show the claim for intervals in [0, t]. Let 0 ≤ a ≤ b ≤ t.We then findµ(t, ω)[a, b] = A(b, ω) − A(a, ω),and by the adaptedness of A, this is F t measurable. Since X is progressively measurable,the extended Fubini Theorem then yields that(X · A) t (ω) =∫ t0X s (ω) dµ(t, ω)(s)is F t measurable as a function of ω, so X · A is adapted.To extend this to the local case, assume X ∈ L 1 (A) and let τ n be a localising sequence.Then, by the ordinary properties of the integral, (X · A) τn = (X[0, τ n ] · A). Therefore,(X · A) τn is adapted, continuous and of finite variation. Clearly, then X · A is adaptedas well, and by Lemma 3.4.1 it is also continuous and of finite variation.Lemma 3.4.4. L 1 (A) is a linear space, and the integral I A : L 1 (A) → cFV is linear.Furthermore, if τ is any stopping time, X[0, τ] ∈ L 1 (A) and (X · A) τ = X[0, τ] · A.

70 The Itô IntegralProof. Since I A is defined as a pathwise Lebesgue integral, this follows from the resultsof ordinary integration theory.We now want to link the definition of the integral with respect to integrators in cFVtogether with our construction of the stochastic integral for integrators in cM L W . By S,we denote the space of processes of the form A + M, where A ∈ cFV and M ∈ cM L W .We call processes in S standard processes. Before making the obvious definition ofintegrals with respect to processes in S, we need to make sure that the decompositionof such processes into finite variation part and local martingale part is unique.Lemma 3.4.5. If F has finite variation and is continuous, then V F can be writtenas V F (t) = sup ∑ nk=1 |F (t k) − F (t k−1 )|, where the supremum is taken over partitionswith rational timepoints.Proof. It will suffice to prove that for any ε > 0 and any partition (t 0 , . . . , t n ), thereexists another partition (q 0 , . . . , q n ) such thatn∑n∑|F (t k ) − F (t k−1 )| − |F (q k ) − F (q k−1 )|∣∣ < ε.k=1ε2nk=1To this end, choose δ parrying for the continuity of F in t 0, . . . , t n , and let, fork ≤ n, q k be some rational with |q k − t k | ≤ δ. Then|(F (t k ) − F (t k−1 )) − (F (q k ) − F (q k−1 ))| ≤ ε n ,and since | · | is a contraction, this implies||F (t k ) − F (t k−1 )| − |F (q k ) − F (q k−1 )|| ≤ ε n ,finally yielding≤n∑|F (t k ) − F (t k−1 )| −∣k=1n∑|F (q k ) − F (q k−1 )|∣k=1n∑||F (t k ) − F (t k−1 )| − |F (q k ) − F (q k−1 )||k=1≤ ε.Lemma 3.4.6. Let A ∈ cFV. Then V A is continuous and adapted.

3.4 Integration with respect to standard processes 71Proof. From Lemma A.1.1, we know that V A is continuous. To show that it is adapted,note that for any partition of [0, t] we have ∑ nk=1 |A t k− A tk−1 | ∈ F t . By Lemma 3.4.5,V A (t) can be written as the supremum of the above sums for partitions of [0, t] withrational timepoints. Since there are only countably many of these, it follows thatV A (t) ∈ F t .Lemma 3.4.7. Let M ∈ cM L 0 . If M has paths of finite variation, then M is evanescent.Proof. We first prove the lemma in a particularly nice setting and then use localisationto get the general result.Step 1: Bounded variation and square-integrability. First assume that V M isbounded and that M ∈ cM 2 0. Then, for any partition (t 0 , . . . , t n ) of [0, t] we obtainby Lemma 2.4.12,EM 2 t = E≤≤n∑(M tk − M tk−1 ) 2k=1E sup |M tk − M tk−1 |k≤nn∑|M tk − M tk−1 |k=1‖V M (t)‖ ∞ E sup |M tk − M tk−1 |.k≤nNow, by the continuity of M, sup k≤n |M tk − M tk−1 | converges almost surely to zero asthe partition grows finer. Noting that sup k≤n |M tk − M tk−1 | is bounded by ‖V M (t)‖ ∞ ,dominated convergence yields EMt2 = 0, so M t is almost surely zero. By continuity,M evanescent.Step 2: Bounded variation. Now assume only that V M (t) is bounded. Sincewe have M ∈ cM L 0 , M is locally in cM 2 0. Let τ n be a localising sequence. ThenV M τn (t) ≤ V M (t), so the previous step yields that M τn is evanescent. Letting n tendto infinity, we obtain that M is evanescent.Step 3: The general case. Finally, merely assume that M ∈ cM L 0 with finitevariation. By Lemma 3.4.6, V M is continuous and adapted. Therefore, by Lemma2.3.5, τ n defined by τ n = inf{t ≥ 0|V M (t) ≥ n} is a stopping time. Since V M isbounded on compact intervals, τ n tends to infinity. M τ n∈ cM L 0 and V M τn (t) isbounded for any t ≥ 0. By what was already shown, we find that M τ nis evanescent.We conclude that M is evanescent.

72 The Itô IntegralLemma 3.4.8. Assume X = A+M, where A has finite variation and M ∈ cM L 0 . Thedecomposition of X into a finite variation process and a continuous local martingaleis unique up to indistinguishability.Proof. Assume that we have another decomposition X = B + N. Then it holds thatA − B = N − M. Therefore, the process N − M is a continuous local martingale withpaths of finite variation. By Lemma 3.4.7, N − M is evanescent. Therefore, A − Bis also evanescent. In other words, A and B are indistinguishable, and so are M andN.We are now finally ready to define the integral with respect to integrators in S. LetX ∈ S with decomposition X = A + M. We put L(X) = L 1 (A) ∩ L 2 (M). We thensimply define , for Y ∈ L(X),Y · X = Y · A + Y · M.This is well-defined up to indistinguishability, since if X = B + N is another decomposition,then Y · A and Y · B are indistinguishable since A and B are indistinguishable,and Y · M and Y · N are indistinguishable since M and N are indistinguishable.Lemma 3.4.9. Let X ∈ S with X = A + M. If Y ∈ L(X), then Y · X ∈ S and hascanonical decomposition given by Y · X = Y · A + Y · M.Proof. This follows immediately from the fact that Y · A ∈ cFV by Lemma 3.4.3 andthe fact that Y · M ∈ cM L 0 by Definition 3.3.8.Lemma 3.4.10. L(X) and S are linear spaces. L(X) is L 0 -stable and S is L-stable.Proof. By Lemma 3.4.1 and Lemma 3.3.11, cFV and cM L W are both L-stable linearspaces. Therefore, S is also a L-stable linear space. Likewise, by Lemma 3.4.4 andLemma 3.3.11, L 1 (A) and L 2 (M) are both L 0 -stable linear spaces, and therefore L(X)is a L 0 -stable linear space.Lemma 3.4.11. The stochastic integral I X : L(X) → S is linear. If τ is a stoppingtime, (Y · X) τ = Y [0, τ] · X = Y · X τ .Proof. The conclusion is well-defined by Lemma 3.4.10. Since we have definedY · X = Y · A + Y · M,

3.4 Integration with respect to standard processes 73the linearity follows from the linearity of the integration with respect to A and M.Likewise, the stopping properties follow from the stopping properties with respect toA and M, shown in Lemma 3.4.4 and Lemma 3.3.11.Finally, we show that the integral defined for integrators X ∈ S and integrals in L(X)cannot be extended further by localisation, in other words, we show that we do notobtain larger classes of integrators and integrands by localising once more.Lemma 3.4.12. Let X ∈ S with decomposition X = A + M. Then, for the spacesof integrators it holds that L(cM L W ) = cML W , L(cFV) = cFV and L(S) = S. Also,for the spaces of integrands, we have L 0 (L(X)) = L(X), L 0 (L 2 (M)) = L 2 (M) andL 0 (L 1 (A)) = L 1 (A).Proof. This follows from Lemma 2.5.13.This concludes our construction of the stochastic integral. We have defined the stochasticintegral for any X ∈ S and Y ∈ L(X) by putting Y · X = Y · A + Y · M, whereX = A + M is the canonical decomposition. We have shown that the integral is linearand interacts properly with stopping times. Further properties of the integral follow byconsidering the components Y ·A and Y ·M and using respectively ordinary integrationtheory and the results from Section 3.3.Before proceeding to the next section, we prove a few useful properties of the stochasticintegral.Lemma 3.4.13. Let X ∈ S. If Y if progressively measurable and locally bounded,then Y ∈ L(X).Proof. Let X = A + M be the canonical decomposition, and let τ n be a localisingsequence such that M τ n∈ cM 2 W . Let σ n be a localising sequence for Y . By ordinaryintegration theory, Y [0, σ n ] ∈ L 1 (A). Since µ M τ k is bounded, Y [0, σ n ] ∈ L 2 (M τ k).Therefore, Y [0, σ n ][0, τ k ] ∈ L 2 (M), so Y [0, σ n ] ∈ L 2 (M). In other words, Y is locallyin L 2 (M), so by Lemma 3.4.12, Y ∈ L 2 (M). We conclude Y ∈ L(X).Next, we prove a result that shows that in a weak sense, the integral with respect to astandard process is consistent with the conventional concept of integration as a limitof Riemann sums. First, we need a lemma.

74 The Itô IntegralLemma 3.4.14. Let X n be a sequence of processes and let τ k be a sequence of stoppingtimes tending to infinity. Let X be another process, and let t ≥ 0 be given. Assumethat X n and X are constant from t and onwards. If for any k ≥ 1, X n (τ k ) convergesto X(τ k ) as n → ∞, either almost surely or in L 2 , then X n (t) converges to X(t) inprobability.Proof. Whether the convergence of X n (τ k ) is almost sure or in L 2 , we have convergencein probability. To show convergence in probability of X n (t) to X(t), let ε > 0 andη > 0 be given. Choose N so large that P (τ k ≤ t) ≤ η for k ≥ N. For such k, we findTherefore,P (|X n (t) − X(t)| > ε) ≤ P (|X n (t) − X(t)| > ε, τ k > t) + P (τ k ≤ t)≤ P (|X n (τ k ) − X(τ k )| > ε) + η.lim sup P (|X n (t) − X(t)| > ε) ≤ η,nand since η was arbitrary, this shows the result.The usefulness of Lemma 3.4.14 may not be clear at present, but we will use it severaltimes in the following, including the very next result to be proven. To set the scene,let 0 ≤ s ≤ t, we say that π = (t 0 , . . . , t n ) is a partition of [s, t] if s = t 0 < · · · < t n = t.We define the norm of the norm ‖π‖ of the partition by ‖π‖ = max k≤n |t k − t k−1 |.Let (X π ) be a sequence of variables indexed by the set of partitions of [0, t], and letX be another variable. We say that X π converges to X in d as the norm of thepartitions tends to zero, or as the mesh tends to zero, if it holds that for any sequenceof partitions π n with norm tending to zero that X π ntends to X in d. Here, d representssome convergence concept, usually L 2 convergence or convergence in probability. Ingeneral, we will attept to formulate our results in terms of convergence as the meshtends to zero instead of using particular sequences of partitions, as the notation for suchpartitions easily becomes cumbersome. An alternative solution would be to considernets instead of sequences. See Appendix C.1 for more on this approach.Lemma 3.4.15. Let X ∈ S and let Y be a bounded, adapted and continuous process.Then Y ∈ L(X), and the Riemann sumsn∑Y tk−1 (X tk − X tk−1 )k=1converge in probability to ∫ t0 Y s dX s as the mesh of the partition tends to zero.

3.4 Integration with respect to standard processes 75Proof. Let X = A + M be the canonical decomposition. Y ∈ L 1 (A) by Lemma 3.4.13.We need to show that for any sequence of partitions with norm tending to zero, theRiemann sums converge to the stochastic integral. Clearly, for any partition,n∑Y tk−1 (X tk − X tk−1 ) =k=1n∑Y tk−1 (A tk − A tk−1 ) +k=1n∑Y tk−1 (M tk − M tk−1 ).We will show that the first term converges to ∫ t0 Y s dA s and that the second termconverges to ∫ t0 Y s dM s . By Lemma A.1.5, the first term converges almost surelyto ∫ t0 Y s dA s , therefore also in probability, when the norm of the partitions tendsto zero. Next, consider the martingale part. First assume that M ∈ cM 2 W . Letπ n = (t n 0 , . . . , t n m n) be given such that ‖π n ‖ tends to zero and put∑m nYs n = Y t nk−11 (t nk−1 ,t k ](s).k=1Then Y n ∈ bE, and by continuity, Y n converges pointwise to Y [0, t]. Since Y isbounded and µ M is bounded, dominated convergence yields that lim ‖Y n − Y ‖ M = 0.By the Itô isometry, we conclude∑m nY t nk−1(M t nk− M t nk−1) =k=1∫ t0Y ns dM sL 2−→∫ t0k=1Y s [0, t] dM s =∫ t0Y s dM s .This shows the claim in the case where M ∈ cM 2 W . Now, if M ∈ cML W , by Lemma3.3.19 there exists a localising sequence τ n such that M τn ∈ cM 2 W . We then haven∑k=1Y tk−1 (M τ nt k− M τ nt k−1) −→L2∫ t0Y s dM τ nsas the mesh tends to zero. Lemma 3.4.14 then allows us to concluden∑∫ tY tk−1 (M tk − M tk−1 ) −→PY s dM s ,as desired.k=10Comment 3.4.16 Let us take a moment to see how exactly the somewhat abstractLemma 3.4.14 was used in the above proof, as we shall often use it again in the samemanner without further details. Our situation is that we know that M ∈ cM L W withM τ n∈ cM 2 W , and n ∑k=1Y tk−1 (M τnt k− Mt τnk−1) −→L2∫ t0Y s dM τ ns

76 The Itô Integralas the mesh tends to zero. Let π n = (t n 0 , . . . , t n m n) be some sequence of partitions withnorm tending to zero. We want to prove∑m nY t nk−1(M t nk− M t nk−1) −→Pk=1∫ t0Y s dM s .Define X n (u) = ∑ m nk=1 Y t n k−1 (M u t n k −Mu t n k−1 ) and X(u) = ∫ t0 Y s dM u s . Since all partitionsare over [0, t], X n and X are constant from t onwards. And by what we already haveshown, X n (τ k ) converges in L 2 to X(τ k ) for any k ≥ 1. Lemma 3.4.14 may now beinvoked to conclude∑m nY t nk−1(M t nk− M t nk−1) = X n (t) −→ PX(t) =k=1∫ t0Y s dM s ,as desired. Since the sequence of partitions was arbitrary, we conclude that we haveconvergence as the mesh tends to zero.Obviously, then, the use of Lemma 3.4.14 in the proof of Lemma 3.4.15 included agood deal of implicit calculations, simple as they may be. It is obvious that if we hadto go through all the details of taking out subsequences, applying Lemma 3.4.14, andgoing back to general partitions, our proofs would become very cluttered. We willtherefore not go through all these diversions when using the lemma. All of the implicitdetails in such cases, however, are completely analogous to the ones that we have gonethrough above.◦Now that we have the definition of the integral in place, we begin the task of developingthe basic results of the stochastic integral. We will prove Itô’s formula, the martingalerepresentation theorem and Girsanov’s theorem. In order to do all this, we need todevelop one of the basic tools of stochastic calculus, the quadratic variation process.This is the topic of the next section.3.5 The Quadratic Variationif F and G are two mappings from [0, ∞) to R, we say that F and G has quadraticcovariation Q : [0, ∞) → R iflimn∑(F (t k ) − F (t k−1 ))(G(t k ) − G(t k−1 )) = Q(t)k=1

3.5 The Quadratic Variation 77for all t ≥ 0 as the mesh of the partitions tend to zero. The quadratic covariation ofF with itself is called the quadratic variation of F . We will now investigate a conceptmuch like this, but instead of real mappings, we will use standard processes. Thus,the basic point of this subsection is to understand the nature of sums of the formn∑(X tk − X tk−1 )(X t ′ k− X t ′ k−1)k=1where X and X ′ are standard processes, when the mesh of the partition tends to zero.Apart from being precisely the kind of variables that are necessary to consider in theproof of Itô’s formula, it also turns out that the resulting quadratic covariation processis a tool of general applicability in the theory of stochastic calculus. We will show thatif X, X ′ ∈ S with decompositions X = A + M and X ′ = A ′ + M ′ , where M = Y · Wand M ′ = Y ′ · W , Y, Y ′ ∈ L 2 (W ) n , thenn∑(X tk − X tk−1 )(X t ′ k− X t ′ k) −→Pk=1n∑k=1∫ t0Y ks (Y ′s) k dsas the mesh becomes finer and finer. For this reason, we will already from now oncall ∑ n∫ tk=1 0 Y s k (Y s) ′ k ds the quadratic covariation of X and X ′ , and we will use thenotation [X, X ′ ] t = ∑ n∫ tk=1 0 Y s k (Y s) ′ k ds. Note that the quadratic covariation doesnot depend on the finite variation components of X and X ′ at all. In particular,[X, X ′ ] = [M, M ′ ]. Also note that we always have [X, X ′ ] ∈ cFV 0 .We write [X] = [X, X] and call [X] the quadratic variation of X. Our strategy fordeveloping the results of this section is first to develop all the necessary results forthe quadratic variation process and then use a concept from Hilbert space theory,polarization, to obtain the corresponding results for the quadratic covariation.Thus, at first, we will forget all about the quadratic covariation and consider only thequadratic variation. We begin by demonstrating some fundamental properties of thequadratic variation process. Our first lemma shows how to calculate the quadraticvariation of a stochastic integral with respect to a martingale integrator. Note thatwe are here using that the quadratic variation is a standard process and can thereforeby used as an integrator.Lemma 3.5.1. Let M ∈ cM L W . Then X ∈ L2 (M) if and only if X ∈ L 2 ([M]).Likewise, X ∈ L 2 (M) if and only if X ∈ L 2 ([M]). In the affirmative case, the equality[X · M] = X 2 · [M] holds.

78 The Itô IntegralProof. Let M = Y · W , where Y ∈ L 2 (W ) n . We then haven∑Ek=1∫ ∞0X 2 s (Y ks ) 2 ds = E= E∫ ∞ n∑Xs20k=1∫ ∞0X 2 s d[M] s .(Ys k ) 2 dsThe first expression is finite if and only if X ∈ L 2 (M), and the final expression is finiteif and only if X ∈ L 2 ([M]). Therefore, X ∈ L 2 (M) if and only if X ∈ L 2 ([M]). Fromlocalisation, it follows that X ∈ L 2 (M) if and only if X ∈ L 2 ([M]).In order to show the integral equality, assume that X ∈ L 2 (M). X · M = XY · W ,and therefore the definition of the quadratic variation yieldsThis shows the lemma.[X · M] t ===n∑∫ tk=10∫ t n∑Xs20k=1∫ t0(X s Y ks ) 2 dsX 2 s d[M] s= (X 2 · [M]) t .(Ys k ) 2 dsOur next two lemmas gives a martingale characterization of the quadratic variationand shows that the quadratic variation interacts properly with stopping.Lemma 3.5.2. Let X be a standard process, and let τ be a stopping time.[X] τ = [X τ ].ThenProof. Let X = A + M with M = Y · W , Y ∈ L 2 (W ) n . X τ then has decompositionX τ = B + N, where B = A τ and N = Y [0, τ] · W , andas desired.[X] τ t =n∑k=1∫ t∧τ0(Y ks ) 2 ds =n∑k=1∫ t0(Y k [0, τ] s ) 2 ds = [X τ ] t ,Lemma 3.5.3. If M ∈ cM L W , [M] is the unique process in cFV 0, up to indistinguishability,that makes M 2 t − [M] t a local martingale. Also, if M ∈ cM 2 W , M 2 t − [M] tis a uniformly integrable martingale.

3.5 The Quadratic Variation 79Proof. We first prove that M 2 t − [M] t is a uniformly integrable martingale when wehave M ∈ cM 2 W . Assume M = Y · W , Y ∈ L2 (W ) n . First note that M has an almostsure limit. Letting τ be any stopping time, we obtainE ( M 2 τ − [M] τ)= E( n∑k=1∫ ∞0Y [0, τ] s dW s) 2−n∑Ek=1∫ ∞0(Y k [0, τ]) 2 s ds = 0by Itô’s isometry, Lemma 3.3.6. Lemma 2.4.13 now yields that Mt 2 −[M] t is a uniformlyintegrable martingale. In the case M ∈ cM L W , letting τ n be a localising sequence, byLemma 3.5.2, [M] τ n= [M τ n] and therefore (M 2 − [M]) τ nt = (M τ n) 2 t − [M τ n] t is amartingale, showing that Mt 2 − [M] t is a local martingale.It remains to show uniqueness. Let A and B be two processes in cFV 0 such thatN A t = M 2 t −A t and N B t = M 2 t −B t both are local martingales. Then N A −N B = B−A,so B − A is a continuous local martingale of finite variation. By Lemma 3.4.7, A andB are indistinguishable.As we will experience in the following, Lemma 3.5.3 is a very important result in thesense that the fact that Mt2 − [M] t is a uniformly integrable martingale wheneverM ∈ cM 2 W is the only fact we will need to identify [M] t as the quadratic variation.In particular, we can in the following completely forget about the specific form of Mas an integral with respect to W . Our next goal will be to show that [X] really canbe interpreted as a quadratic variation, we will show the resultn∑(X tk − X tk−1 ) 2 P−→ [X]t .k=1For any partition π and stochastic processes X and X ′ we define the quadratic covariationof X and X ′ over π by Q π (X, X ′ ) = ∑ nk=1 (X t k−X tk−1 )(X ′ t k−X ′ t k−1). We writeQ π (X) = Q π (X, X). The next two lemmas are technical results that pave the roadfor the result on the convergence of the squared increments Q π (X) to the quadraticvariation.Lemma 3.5.4. If M ∈ cM 2 Wand Y is a bounded and adapted process, it holds thatE= E(∑ nY tk−1 ((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 ))k=1) 2n∑ (Ytk−1 ((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 )) 2) 2,k=1in other words, the cross-terms equal zero.

80 The Itô IntegralProof. Consider i < j with i ≤ n, j ≤ n. Let X i = (M ti − M ti−1 ) 2 − ([M] ti − [M] ti−1 ).Then X i ∈ F ti ⊆ F tj−1 , so we obtainEY ti−1 X i Y tj−1 X j = E(Y ti−1 X i Y tj−1 E(X j |F tj−1 ).Now noteE(X j |F tj−1 ) = E((M tj − M tj−1 ) 2 − ([M] tj − [M] tj−1 )|F tj−1 )= E(Mt 2 j− [M] tj + Mt 2 j−1+ [M] tj−1 − 2M tj M tj−1 |F tj−1 )= E(Mt 2 j− [M] tj − (Mt 2 j−1− [M] tj−1 )|F tj−1 )= 0,by Lemma 3.5.3. Therefore, we findas desired.( n) 2∑E Y tk−1 ((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 )) 2k=1( n) 2∑= E Y tk−1 X k== E= En∑k=1k=1 i=1n∑EY tk−1 X k Y ti−1 X in∑k=1Y 2t k−1X 2 kn∑ (Ytk−1 ((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 )) 2) 2,k=1Lemma 3.5.5. Let M be a continous bounded martingale, zero at zero. Thenn∑E (M tk − M tk−1 ) 4k=1tends to zero as the norm of the partition tends to zero.Proof. We first employ the Cauchy-Schwartz inequality to obtainn∑k=1∑ nE(M tk − M tk−1 ) 4 ≤ E sup(M tk − M tk−1 ) 2 (M tk − M tk−1 ) 2k≤nk=1() 1≤ E sup(M tk − M tk−1 ) 4 2 (EQ π (M) 2) 1 2.k≤n

3.5 The Quadratic Variation 81The first factor converges to zero by dominated convergence as ‖π‖ tends to zero, sinceM is continuous and bounded. It is therefore sufficient for our needs to show that thesecond factor is bounded. We rewrite asEQ π (M) 2( n) 2∑= E (M tk − M tk−1 ) 2=n∑k=1k=1n−1∑E(M tk − M tk−1 ) 4 + 2n∑i=1 j=i+1The last term can be computed using Lemma 2.4.12 twice,n−1∑2n∑i=1 j=i+1n−1∑= 2 Ei=1⎛E(M ti − M ti−1 ) 2 (M tj − M tj−1 ) 2⎛⎝(M ti − M ti−1 ) 2 E ⎝E(M ti − M ti−1 ) 2 (M tj − M tj−1 ) 2 .∣ ⎞⎞n∑∣∣∣∣∣(M tj − M tj−1 ) 2 F ti⎠⎠j=i+1n−1∑= 2 E ( (M ti − M ti−1 ) 2 E((M t − M ti ) 2 |F ti ) )≤i=1n−1∑8‖M ∗ ‖ 2 ∞ E(M ti − M ti−1 ) 2i=1≤ 8‖M ∗ ‖ 2 ∞EM 2 t .We then find, again using Lemma 2.4.12,EQ π (M) 2≤≤≤n∑E(M tk − M tk−1 ) 4 + 8‖M ∗ ‖ 2 ∞EMt2k=14‖M ∗ ‖ 2 ∞n∑E(M tk − M tk−1 ) 2 + 8‖M ∗ ‖ 2 ∞EMt2k=14‖M ∗ ‖ 2 ∞EM 2 t + 8‖M ∗ ‖ 2 ∞EM 2 t= 12‖M ∗ ‖ 2 ∞EM 2 t ,showing the desired boundedness.Finally, we are ready to show that the quadratic variation can actually be interpretedas a quadratic variation. We begin with the martingale case.Theorem 3.5.6. Let M ∈ cM 2 W . Assume that M and [M] are bounded. With π apartition of [0, t], Q π (M) tends to [M] t in L 2 as ‖π‖ tends to zero.

82 The Itô IntegralProof. By Lemma 3.5.4, using (x − y) 2 ≤ 2x 2 + 2y 2 ,(∑ nE (Q π (M) − [M] t ) 2 = E (M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 )=k=1n∑E((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 )) 2k=1n∑n∑≤ 2 E(M tk − M tk−1 ) 4 + 2 E([M] tk − [M] tk−1 ) 2 .k=1To show the result of the theorem, it therefore suffices to show that each of the twosums above converges to zero as ‖π‖ tends to zero. Now, the first sum converges tozero by 3.5.5. Concerning the second sum, we have, since [M] is increasing,n∑E ( ) 2[M] tk − [M] tk−1 ≤ ∣ ∑E sup ∣ n ∣[M]tk − [M] tk−1 ∣ [M]tk − [M] tk−1 k≤nk=1k=1k=1) 2∣∣ ∑≤ E sup ∣[M] n tk − [M] tk−1 [M] tk − [M] tk−1k≤nk=1∣= E[M] t sup ∣ [M]tk − [M] tk−1 ,k≤nwhich tends to zero by dominated convergence as ‖π‖ tends to zero, since [M] iscontinuous and bounded.Corollary 3.5.7. Let M ∈ cM L 0 . Then Q π (M) converges in probability to [M] t as‖π‖ tends to zero.Proof. Let τ n = inf{t ≥ 0|M t > n or [M] t > n}. Then τ n is a stopping time, and τ ntends to infinity because M and [M] are continuous. Furthermore, M τn and [M τn ] arebounded. Thus, the hypotheses of Theorem 3.5.6 are satisfied and Q π (M τ n) convergesin L 2 to ∫ t0 Y [0, τ n] 2 s ds as ‖π‖ tends to zero. Since we know Q π (M τ n) = Q π (M) τ nand [M] τ n= [M τ n], the result now follows from Lemma 3.4.14.Theorem 3.5.8. Let X ∈ S. With π a partition of [0, t], Q π (X) tends to [X] t inprobability as ‖π‖ tends to zero.

3.5 The Quadratic Variation 83Proof. Letting X = A + M, we first noten∑(X tk − X tk−1 ) 2 =k=1=+ 2+n∑(A tk − A tk−1 + M tk − M tk−1 ) 2k=1n∑(A tk − A tk−1 ) 2k=1n∑(A tk − A tk−1 )(M tk − M tk−1 )k=1n∑(M tk − M tk−1 ) 2 .For the first term, we obtain∣ n∑∣∣∣∣ (A∣tk − A tk−1 ) 2 ≤ max |A t k− A tk−1 |V A (t),k≤nk=1k=1and since A is continuous, the above tends to zero almost surely. For the second term,we find ∣ ∣∣∣∣ ∑n (A tk − A tk−1 )(M tk − M tk−1 )∣ ≤ max |M t k− M tk−1 |V A (t),k≤nk=1and again conclude that this tends to zero almost surely, by the continuity of M. Andfrom Corollary 3.5.7, we know that the last term tends to [M] t , which is equal to[X] t .Theorem 3.5.8 shows that the process [X] really is a quadratic variation in the sensethat it is a limit of quadratic increments. It remains to show that [X, X ′ ] has theanalogous interpretation of a quadratic covariation. Before this, we show a simplegeneralisation of Theorem 3.5.8 which will be useful to us in the proof of Itô’s formula.It is the analogue to Lemma 3.4.15, using quadratic Riemann sums instead of ordinaryRiemann sums.Corollary 3.5.9. Let X ∈ S. Let Y be a bounded, adapted and continuous process.Then the quadratic Riemann sumsn∑Y tk−1 (X tk − X tk−1 ) 2k=1converge in probability to ∫ t0 Y s d[X] s as the mesh of the partition tends to zero.

84 The Itô IntegralProof. Letting X = A + M, we haven∑n∑Y tk−1 (X tk − X tk−1 ) 2 = Y tk−1 (A tk − A tk−1 + M tk − M tk−1 ) 2k=1=+ 2+k=1n∑Y tk−1 (A tk − A tk−1 ) 2k=1n∑Y tk−1 (A tk − A tk−1 )(M tk − M tk−1 )k=1n∑Y tk−1 (M tk − M tk−1 ) 2 .k=1For the first term, we obtain∣ n∑∣∣∣∣ Y∣tk−1 (A tk − A tk−1 ) 2 ≤ ‖Y ‖ ∞ max |A t k− A tk−1 |V A (t),k≤nk=1and since A is continuous, the above tends to zero almost surely. For the second term,we findn∑Y∣tk−1 (A tk − A tk−1 )(M tk − M tk−1 )∣ ≤ ‖Y ‖ ∞ max |M t k− M tk−1 |V A (t),k≤nk=1and again conclude that this tends to zero almost surely, by the continuity of M. Therefore,it will suffice to show that the final term converges in probability to ∫ t0 Y s d[X] s .Recall here that [X] = [M], so we really have to show convergence to ∫ t0 Y s d[M] s .We first consider the case where M and [M] are bounded. By Lemma 3.4.15,n∑Y s ([M] tk − [M] tk−1 ) −→Pk=1∫ t0Y s d[M] s ,so it will suffice to show that ∑ nk=1 Y s((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 )) tends to

3.5 The Quadratic Variation 85zero in probability. But we have, by Lemma 3.5.4,= E≤(∑ nE Y s ((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 ))k=1n∑(Y s ((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 ))) 2k=1‖Y ∗ ‖ 2 ∞E) 2n∑((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 )) 2k=1(∑ n= ‖Y ∗ ‖ 2 ∞E ((M tk − M tk−1 ) 2 − ([M] tk − [M] tk−1 ))k=1= ‖Y ∗ ‖ 2 ∞E (Q π (M) − [M] t ) 2 ,which converges to zero by Theorem 3.5.6. Now consider the general case whereM ∈ cM L W . Let τ n be a localising sequence such that M τn and [M] τn are bounded, thiscan be done since [M] and M both are continuous. By Lemma 3.5.2, [M] τn = [M τn ],and we then obtainn∑k=1Y tk−1 (M τ nt kThe result then follows from Lemma 3.4.14.∫ t∧τn− M τ nt k−1) 2 P−→0Y s d[X] s .) 2We now transfer the results on the quadratic variation to the setting of the quadraticcovariation. Recall that we defined [X, X ′ ] t = ∫ t0 Y sY s ′ ds. As mentioned earlier, themethod to do so is called polarization. The name originates from the polarizationidentity for inner product spaces,〈x, y〉 = 1 4(‖x + y‖ 2 − ‖x − y‖ 2) ,see Lemma 11.2 of Meise & Vogt (1997). The polarization identity allows one torecover the inner product from the norm. Thinking of the quadratic variation as thesquare norm and of the quadratic covariation as the inner product, we will show thatthe two satisfies a relation analogous to the above, and we will use this relation toobtain results on the quadratic covariation from the quadratic variation.Lemma 3.5.10. Let X, X ′ ∈ S. Then[X, X ′ ] = 1 4 ([X + X′ ] − [X − X ′ ]) .

86 The Itô IntegralProof. Assume that X = A+M and X ′ = A ′ +M ′ , where M = Y ·W and M ′ = Y ′·W .Therefore,14 ([X + X′ ] t − [X − X ′ ] t ) = 1 (∫ t∫ t)(Y s + Y s ′ ) 2 ds − (Y s − Y s ′ ) 2 ds40∫ t= 1 4 0= [X, X ′ ].4Y s Y ′s ds0Lemma 3.5.11. Let M, N ∈ cM L W . Then [M, N] is the unique process in cFV 0 suchthat MN − [M, N] is a local martingale.Proof. That MN − [M, N] is a local martingale follows from Lemma 3.5.3, sinceMN − [M, N] = 1 4 ((M + N)2 − (M − N) 2 ) − 1 ([M + N] − [M − N])4= 1 4 ((M + N)2 − [M + N]) − 1 4 ((M − N)2 − [M − N]).Uniqueness follows as in Lemma 3.5.3 from Lemma 3.4.7.Theorem 3.5.12. Let X, X ′ ∈ S and let Y be a bounded, adapted and continuousprocess. The quadratic Riemann sumsn∑Y tk−1 (X tk − X tk−1 )(X t ′ k− X t ′ k−1)k=1tend to ∫ t0 Y s d[X, X ′ ] s as the mesh of the partitions tend to zero.Proof. We find thatn∑Y tk−1 (X tk − X tk−1 )(X t ′ k− X t ′ k−1)k=1= 1 4− 1 4n∑Y tk−1 (X tk + X t ′ k− (X tk−1 + X t ′ k−1)) 2k=1n∑Y tk−1 (X tk − X t ′ k− (X tk−1 − X t ′ k−1)) 2 .k=1∫By Corollary 3.5.9, the first term converges in probability to 1 t4 0 Y s d[X + X ′ ] s and∫the second term converges in probability to 1 t4 0 Y s d[X − X ′ ] s , as the mesh of the

3.5 The Quadratic Variation 87partitions tend to zero. Therefore, ∑ nk=1 Y t k−1(X tk − X tk−1 )(X t ′ k− X t ′ k−1) convergesin probability to14∫ t0Y s d[X + X ′ ] s + 1 4by Lemma 3.5.10.∫ t0Y s d[X − X ′ ] s ==∫ t0∫ t0( 1Y s d4 [X + X′ ] s + 1 )4 [X − X′ ] sY s d[X, X ′ ] sThe main results on the quadratic covariation of this section are Lemma 3.5.11 andTheorem 3.5.12. Putting Y = 1 in Theorem 3.5.12 shows that our definition of thequadratic covariation of two standard processes really can be interpreted as a quadraticcovariation, although the convergence is only in probability.Before concluding, we note an important consequence of the definition of the quadraticvariation. Since the quadratic variation is a continuous and adapted process of finitevariation, it is a standard process, and we can therefore integrate with respect to it.The following lemma exploits this feature.Lemma 3.5.13. Let X, X ′ ∈ S and assume that Y ∈ L 1 (X) and Y ′ ∈ L 1 (X ′ ). ThenY Y ′ ∈ L 1 ([X, X ′ ]) and[Y · X, Y ′ · X ′ ] = Y Y ′ · [X, X ′ ].Proof. Let the canonical decompositions be X = A + M and X ′ = A ′ + M ′ . Then[Y · X, Y ′ · X ′ ] = [Y · M, Y ′ · M ′ ] and [X, X ′ ] = Y Y ′ · [M, M ′ ], so it will suffice to showthe result in the case where the finite variation components are zero. The integrabilityresult follows from the definition of the quadratic covariation and results from measuretheory. For the second statement of the lemma, we calculate[Y · M, Y ′ · M ′ ] t =n∑k=1∫ t0Y s Z k s Y ′s(Z ′ s) k ds =∫ t0Y s Y ′s d[M, M ′ ] s = (Y Y ′ · [M, M ′ ]) s .Lemma 3.5.11 provides a characterisation of the quadratic covariation only in termsof martingales. This result shows that quadratic covariation is essentially a purelymartingale concept, even though we here have developed it in the context of stochasticintegration. This observation is the key to extending the stochastic integral to using

88 The Itô Integralgeneral continuous local martingales as integrators. The price of this generality is thatin a purely martingale setting, we do not have any obvious candidate for the quadraticcovariation. In our setting, the existence of the quadratic variation was a trivialityonce the correct idea had been identified. In general, the proof of the existence ofthe quadratic covariation is quite difficult. In Rogers & Williams (2000b), Theorem30.1, a direct construction of the quadratic variation of a continuous square-integrablemartingale is carried out. Alternatively, one can obtain the existence in a more abstractmanner by invoking the Doob-Meyer decomposition theorem, as is done in Karatzas& Shreve (1988). Another alternative is presented in Appendix C.1, where we give ageneral existence result on the quadratic variation in the continuous case based onlyon relatively elementary martingale theory.3.6 Itô’s FormulaIn this section, we will prove the multidimensional Itô formula. We say that X is ann-dimensional standard process if each of X’s coordinates is a standard process. TheItô formula states that if X is a n-dimensional standard process and f ∈ C 2 (R n ), thenf(X t ) = f(X 0 ) +n∑i=1∫ t0∂f∂x i(X s ) dX i s + 1 2n∑n∑i=1 j=1∫ t0∂ 2 f∂x i ∂x j(X s ) d[X i , X j ] s .This important result in particular shows that a C 2 transformation of a standard processis again a standard process, and allows us to identify the canonical decomposition.To prove this result, we will first localise to a particularly nice setting, use a Taylorexpansion of f and take limits. Before starting the proof, we prepare ourselves withthe right version of the Taylor expansion theorem and a result that will help us whenlocalising in the proof.Theorem 3.6.1. Let f ∈ C 2 (R n ), and let x, y ∈ R n . There exists a point ξ ∈ R n onthe line segment between x and y such thatf(y) = f(x) +n∑i=1∂f∂x i(x)(y i − x i ) + 1 2n∑n∑i=1 j=1∂ 2 f∂x i ∂x j(ξ)(y i − x i )(y j − x j ).Proof. Define g : R → R by g(t) = f(x + t(y − x)). Note that g(1) = f(y) andg(0) = f(x). We will prove the theorem by applying the one-dimensional Taylorformula, see Apostol (1964) Theorem 7.6, to g. Clearly, g ∈ C 2 (R), and we obtain

3.6 Itô’s Formula 89g(1) = g(0) + g ′ (0) + 1 2 g′′ (s), where 0 ≤ s ≤ 1. Applying the chain rule, we findandg ′′ (t) =g ′ (t) =n∑n∑i=1 j=1n∑i=1∂f∂x i(x + t(y − x))(y i − x i )∂ 2 f∂x i ∂x j(x + t(y − x))(y i − x i )(y j − x j ).Substituting and writing ξ = x + s(y − x), we may concludef(y) = f(x) +n∑i=1∂f∂x i(x)(y i − x i ) + 1 2n∑n∑i=1 j=1∂ 2 f∂x i ∂x j(ξ)(y i − x i )(y j − x j ).Lemma 3.6.2. Let f ∈ C 2 (R n ). For any compact set K, There exists g ∈ C 2 c (R n )such that f and g are equal on K.Proof. By Urysohn’s Lemma, Theorem A.3.4, there exists ψ ∈ C ∞ c (R n ) with K ≺ ψ.Defining g(x) = ψ(x)f(x), g ∈ C 2 c (R n ) and f and g are clearly equal on K.Theorem 3.6.3. Let X be a n-dimensional standard process, and let f ∈ C 2 (R n ).Thenf(X t ) = f(X 0 ) +n∑i=1up to indistinguishability.∫ t0∂f∂x i(X s ) dX i s + 1 2n∑n∑i=1 j=1∫ t0∂ 2 f∂x i ∂x j(X s ) d[X i , X j ] s ,Proof. First note that all of the integrands are continous, therefore locally boundedand therefore, by Lemma 3.4.13 the stochastic integrals are well-defined.Our plan is first to reduce by localization to a particularly nice setting, apply theTaylor formula to obtain an expression for f(X t ) − f(X 0 ) in terms of Riemann-likesums and then use Lemma 3.4.15 and Theorem 3.5.12 on convergence of Riemann andquadratic Riemann sums.Step 1: The localized case. We first consider the case where X is bounded. Thevariables f(X t ), ∂f∂x i(X t ) and∂2 f∂x i ∂x j(X t ) does not depend on the values of f outside therange of X. Therefore, we can by Lemma 3.6.2 assume that f has compact support. In

90 The Itô Integral∂fthat case, all of f,∂x iand ∂2 f∂x i∂x jare bounded. Consider a partition π = (t 0 , . . . , t n )of [0, t]. The Taylor Formula 3.6.1 with Lagrange’s remainder term then yields==+ 1 2f(X t ) − f(X 0 )n∑f(X tk ) − f(x tk−1 )k=1n∑n∑i=1 k=1n∑∂f∂x i(X tk−1 )(X i t k− X i t k−1)n∑n∑i=1 j=1 k=1∂x i∂ 2 f∂x i ∂x j(ξ k )(X i t k− X i t k−1)(X j t k− X j t k−1).We will investigate the limit of the sums as the mesh of the partition tends to zero.By Lemma 3.4.15, since ∂f is bounded, ∑ n ∂fk=1 ∂x i(X tk−1 )(Xt i k− Xt i k−1) converges inprobability to ∫ t0n∑i=1 k=1∂f∂x i(X s ) dX i s as the mesh of the partitions tend to zero. Therefore,n∑∂f∂x i(X tk−1 )(X i t k− X i t k−1)P−→n∑i=1∫ t0∂f∂x i(X s ) dX i s.It remains to consider the second-order terms. Fix i, j ≤ n. We first note that thedifference between the sums given by ∑ n ∂ 2 fk=1 ∂x i ∂x j(ξ k )(Xt i k− Xt i k−1)(X j t k− X j t k−1) and∑ n ∂ 2 fk=1 ∂x i ∂x j(X tk−1 )(Xt i k− Xt i k−1)(X j t k− X j t k−1) is bounded bymaxk≤n∂ 2 f∣ (ξ k ) −∂2 fn∑(X tk−1 )∂x i ∂x j ∂x i ∂x j∣ (Xt i k− Xt i k−1)(X j t k− X j t k−1).By continuity of X and f ′′ , the maximum tends to zero almost surely. By Theorem3.5.12, ∑ nk=1 (X t k− X tk−1 ) 2 tends to [X] t in probability. Therefore, the above tendsto zero in probability as the mesh tends to zero. It will therefore suffice to prove thatn∑k=1k=1∂ 2 f∂x i ∂x j(X tk−1 )(X i t k− X i t k−1)(X j t k− X j t k−1)∫ t−→P0∂ 2 f∂x i ∂x j(X s ) d[X i , X j ] s ,as the mesh tends to zero, but this follows immediately from Theorem 3.5.12. We maynow conclude that for each t ≥ 0,f(X t ) = f(X 0 ) +n∑i=1∫ t0∂f∂x i(X s ) dX i s + 1 2n∑n∑i=1 j=1∫ t0∂ 2 f∂x i ∂x j(X s ) d[X i , X j ] s ,almost surely. Since both sides are continuous, we have equality up to indistinguishability.

3.6 Itô’s Formula 91Step 2: The general case. Since each X i is continuous, there exists a sequenceof stopping times τ n tending to infinity such that X τ nis bounded. By what we havealready shown, we then havef(X t ) τ n= f(X τnt )= f(X 0 ) ++ 1 2n∑n∑i=1 j=1= f(X 0 ) ++ 1 2n∑n∑i=1 j=1n∑∫ ti=1 0∫ t0∫ t∧τnn∑i=1 0∫ t∧τn0∂f(X τ ns ) d(X τ n) i s∂x i∂ 2 f(X τ ns ) d[(X τ n) i , (X τ n) j ] s∂x i ∂x j∂f∂x i(X s ) d(X) i s∂ 2 f∂x i ∂x j(X s ) d[X i , X j ] s .The Itô formula therefore holds whenever t ≤ τ n . Since τ n tends to infinity, thetheorem is proven.We have now proven Itô’s formula. This result is more or less the center of the theoryof stochastic integration. We will sometimes need to apply Itô’s formula to mappingswhich are not defined on all of R n . The following corollary shows that this is possible.Corollary 3.6.4. Let X be a n-dimensional standard process taking its values in anopen set U. Let f ∈ C 2 (U). Thenf(X t ) = f(X 0 ) +n∑i=1up to indistinguishability.∫ t0∂f∂x i(X s ) dX i s + 1 2n∑n∑i=1 j=1∫ t0∂ 2 f∂x i ∂x j(X s ) d[X i , X j ] s ,Proof. Define F n = {x ∈ R n |d(x, U c ) ≥ 1 n}. Our plan is to in some sense localise toF n and prove the result there. F n is a closed set with F n ⊆ U. Let g n ∈ C ∞ (R n )be such that F n ≺ g n ≺ Fn+1, c such a mapping exists by Lemma A.3.5. Definef n (x) = g n (x)f(x) when x ∈ U and zero otherwise. Clearly, f n and f agree on F n .We will argue that f n ∈ C 2 (R n ). To this end, note that since f and g n are both C 2 onU, it is clear that f n is C 2 on U. Since f n is zero on Fn+1, c f n is C 2 on Fn+1. c BecauseR n = U c ∪ U ⊆ Fn+1 c ∪ U, this shows f n ∈ C 2 (R n ). Itô’s Lemma then yieldsf n (X t ) = f n (X 0 ) +n∑i=1∫ t0∂f n∂x i(X s ) dX i s + 1 2n∑n∑i=1 j=1∫ t0∂ 2 f n∂x i ∂x j(X s ) d[X i , X j ] s .

92 The Itô IntegralNow define τ n = inf{t ≥ 0|d(X t , U c ) ≤ 1 n }. τ n is a stopping time, and since d(X t , U c )is a positive continuous process, τ n tends to infinity. When t ≤ τ n , d(X t , U c ) < 1 n , soX t ∈ F n . Since f n and f agree on F n , we concludef(X t ) τn= f n (X 0 ) += f(X 0 ) +n∑∫ t∧τni=1 0∫ t∧τnn∑i=10∂f n∂x i(X s ) dX i s + 1 2∂f∂x i(X s ) dX i s + 1 2Letting τ n tend to infinity, we obtain the result.n∑n∑∫ t∧τni=1 j=1 0∫ t∧τnn∑n∑i=1 j=10∂ 2 f n∂x i ∂x j(X s ) d[X i , X j ] s∂ 2 f∂x i ∂x j(X s ) d[X i , X j ] s .3.7 Itô’s Representation TheoremIn this section and the following, we develop some results specifically for the contextwhere we have a n-dimensional Brownian motion W on (Ω, F, P ) and let the filtrationF t be the one induced by W , augmented in the usual manner as described in Section2.1. We will need to work with processes on [0, T ] and integration on [0, T ]. We willtherefore make some notation for this situation. By L 2 T (W ), we denote the spaceL 2 ([0, T ] × Ω, Σ π [0, T ], λ ⊗ P ), understanding that Σ π [0, T ] is the restriction of Σ πto [0, T ] × Ω. The norm on L 2 T (W ) is denoted ‖ · ‖ W T . The integral of a processY ∈ L 2 T (W )n is then considered a process on [0, T ], defined as the integral of theprocess Y [0, T ].Our goal of this section is to prove Itô’s representation theorem, which yields a representationof all cadlag F t local martingales. To prove the result, we will need somepreparation. We begin with a few lemmas.Lemma 3.7.1. Assume that X ∈ S with X 0 = 0 and let Y 0 be a constant. Thereexists precisely one solution to the equation Y t = Y 0 + ∫ t0 Y s dX s in Y , unique up toindistinguishability, and the solution is given by Y t = Y 0 exp(X t − 1 2 [X] t).Proof. We first check that Y as given in the lemma in fact solves the equation. Thecase Y 0 = 0 is trivial, so we assume Y 0 ≠ 0. We note that Y t = f(X t , [X] t ) withf(x, y) = Y 0 exp(x − 1 2 y). Itô’s formula then yields, using that X 0 = 0 and therefore

3.7 Itô’s Representation Theorem 93exp(X 0 − 1 2 [X] 0) = 1,Y t = Y 0 += Y 0 +∫ t0∫ t0Y s dX s − 1 2Y s dX s ,∫ t0Y s d[X] s + 1 2∫ t0Y s d[X] sas desired. Now consider uniqueness. Let Y be any solution of the equation, anddefine Y ′ by Y ′ = exp(−X t + 1 2 [X] t). Itô’s formula yieldsY ′t = Y 0 −= Y 0 −∫ tY ′0∫ t0s dX s + 1 2Y ′s dX s +∫ t0∫ t0Y ′s d[X] s + 1 2Y ′s d[X] s .Therefore, again by Itô’s formula, using Lemma 3.5.13,Y t Y ′t = Y 0 += Y 0 −= Y 0 ,∫ t0∫ t0Y s dY ′s +∫ tY s Y ′s dX s +Y ′s dY s + [Y, Y ′ ] t0∫ t0Y s Y ′s d[X] s +∫ t0∫ t0Y ′s d[X] sY ′sY s dX s −up to indistinguishability. Inserting the expression for Y ′ , we conclude(Y t = Y 0 exp X t − 1 )2 [X] t ,demonstrating uniqueness.∫ t0Y s Y ′s d[X] sThe equation Y t = Y 0 + ∫ t0 dX s of Lemma 3.7.1 is called a stochastic differentialequation, or a SDE. This equation is one of the few SDEs where an explicit solutionis available. This type of equation will play an important role in the following, andwe therefore introduce some special notation for the solution. For any X ∈ S withX 0 = 0, we define the stochastic exponential of X by E(X) t = exp(X t − 1 2 [X] t).The stochastic exponential is also known as the Doléans-Dade exponential. If M isa local martingale with M 0 = 0, we see from E(M) t = 1 + ∫ t0 E(M) t dM t that E(M)also is a local martingale. We will now give a criterion to check when E(M) is asquare-integrable martingale.Lemma 3.7.2. Let M be a nonnegative continuous local martingale. Then M is asupermartingale.

94 The Itô IntegralProof. Let 0 ≤ s ≤ t, and let τ n be a localising sequence.conditional expectations and continuity of M,By Fatou’s Lemma forE(M t |F s ) = E(lim inf M τnt |F s )≥lim inf E(M τnt |F s )= lim inf M τ ns= M s .Lemma 3.7.3. Assume that Y ∈ L 2 (W ) n and that there exists f k ∈ L 2 [0, ∞) suchthat |Y kt | ≤ f k (t) for k ≤ n. Put M = Y · W . Then E(M) ∈ cM 2 0.Proof. From Lemma 3.7.1, we have E(M) tformula yields= 1 + ∫ t0 E(M) s dM s . Therefore, Itô’sE(M) 2 t = 1 + 2= 1 + 2= 1 + 2≤ 1 + 2∫ t0∫ t0∫ t0∫ t0E(M) s dE(M) s + [E(M)] tE(M) 2 s dM s +E(M) 2 s dM s +E(M) 2 s dM s +∫ t0n∑k=1∫ t0E(M) 2 s d[M] s∫ t0E(M) 2 sE(M) 2 s(Y ks ) 2 dsn∑f k (s) 2 ds.Define b(s) = ∑ nk=1 f k(s) 2 . Since ∫ t0 E(M)2 s dM s is a continuous local martingale,there exists a localising sequence τ m such that E(M) 2 t∧τ mis bounded over t and∫ t∧τmE(M) 2 0 s dM s is a bounded martingale in t. We then obtain∫ t∧τmEE(M) 2 t∧τ m≤ 1 + E E(M) 2 sb(s) ds= 1 + E= 1 + E≤1 + E= 1 +0∫ t0∫ t0∫ t0∫ t0k=1E(M) 2 sb(s)1 [0,τm](s) dsE(M) 2 s∧τ mb(s)1 [0,τm ](s) dsE(M) 2 s∧τ mb(s) dsE(E(M) 2 s∧τ m)b(s) ds,

3.7 Itô’s Representation Theorem 95by the Tonelli theorem. Now consider m fixed. Gronwall’s Lemma A.2.1 then yieldsEE(M) 2 t∧τ mUsing Fatou’s Lemma, we then obtain(∫ t)≤ exp b(s) ds .0(∫ t)EE(M) 2 t = E lim infm E(M)2 t∧τ m≤ lim infm EE(M)2 t∧τ m= exp0b(s) ds .Since E(M) is a nonnegative local martingale, it is a supermartingale by Lemma 3.7.2.Therefore, E(M) is almost surely convergent by the Martingale Convergence Theorem2.4.3. For the limit we then obtain, again using Fatou’s Lemma,EE(M) 2 ∞ = E lim inft(∫ ∞)E(M) 2 t ≤ lim inf EE(M) 2 t ≤ expt0b(s) ds < ∞.Therefore, E(M) is bounded in L 2 . This shows the lemma.Our next lemma will require some basic results from the theory of weak convergence.An excellent standard reference to this is Billingsley (1999).Lemma 3.7.4. Let h 1 , . . . , h n ∈ L 2 [0, T ], and let W denote a one-dimensional Brownianmotion. Then the vector ( ∫ T0 h1 s dW s , . . . , ∫ T0 hn s dW s ) follows a n-dimensionalnormal distribution with mean zero and variance matrix Σ given by Σ ij = ∫ T0 hi sh j s ds.Proof. First consider the case where h 1 , . . . , h n are continuous. Let π = (t 0 , . . . , t m )be a partition of [0, T ] and define X π by putting Xiπ = ∑ mk=1 hi (t k−1 )(W tk − W tk−1 ).X π is then a stochastic variable with values in R n . Since X π is a linear transformationof normal variables, X π is normally distributed. It is clear that X π has mean zero,and the covariance matrix Σ π of X π is( m) (∑m)∑Σ π ij = E h i (t k−1 )(W tk − W tk−1 ) h j (t k−1 )(W tk − W tk−1 )=k=1m∑h i (t k−1 )h j (t k−1 )(t k − t k−1 ) 2 .k=1Now, as the mesh tends to zero, Σ π ij tends to ∫ T0 hi sh j s ds. Therefore, X π tends to thenormal distribution with mean zero and the covariance matrix Σ given in the statementof the lemma.k=1

96 The Itô IntegralOn the other hand, for any a ∈ R n , we have by Lemma 3.4.15,n∑a i Xiπi=1P−→n∑∫ Ta i h i (s) dW s .i=10as the mesh tends to zero. Since convergence in probability implies weak convergence,by the Cramér-Wold Device, X π tends to ( ∫ T0 h n(s) dW s , . . . , ∫ T0 h n(s) dW s ) weakly.By uniqueness of limits, we conclude that the lemma holds in the case of continuoush 1 , . . . , h n .Now consider the general case, where h 1 , . . . , h n ∈ L 2 [0, T ]. For i ≤ n, there exists asequence (h i k ) of continuous maps converging in L2 [0, T ] to h i . By what we alreadyhave shown, these are normally distribued with zero mean and covariance matrixΣ k ij = ∫ T0 hi k (s)hj k(s) ds. Since the integral is continuous, the Cramér-Wold Deviceand uniqueness of limits again yields the conclusion.Lemma 3.7.5. The span of the variables(∑ n ∫ Texp h k (t) dWt k − 1 2k=1where h 1 , . . . , h n ∈ L 2 [0, T ], is dense in L 2 (F T ).0n∑k=1∫ T0h 2 k(t) dt)Proof. Let (G t ) denote the filtration generated by W . From the construction of theusual augmentation in Theorem 2.1.4, we know that F T = σ(G T + , N ), where N denotesthe null sets of F. By Lemma 2.1.9, G T + ⊆ σ(G T , N ). We therefore concludeF T ⊆ σ(G T , N ), so it will suffice to show that the span of the variables given in thestatement of the lemma is dense in L 2 (σ(G T , N )). By Lemma B.6.9, it will then sufficeto show that the variables are dense in L 2 (G T ).To this end, first note that ∫ T0 1 [0,s](t) dWt k = Ws k , so G T is generated by the variablesof the form ∑ n∫ Tk=1 0 h k(t) dWt k . Also, by Lemma 3.7.4, the vector of variables∑ n∫ Tk=1 0 h k(t) dWt k is normally distributed, in particular it has exponential momentsof all orders. It then follows from linearity of the integral and Theorem B.6.2 that thevariables exp( ∑ n∫ Tk=1 0 h k(t) dWt k ) are dense in L 2 (G T ). Since 1 ∑ n∫ T2 k=1 0 h2 k(t) dt isconstant, the conclusion of the lemma follows.We are now more or less ready to begin the proof of the martingale representationtheorem. We start out with a version of the theorem for finite time intervals.

3.7 Itô’s Representation Theorem 97Theorem 3.7.6. Let X ∈ L 2 (F T ). There exists Y ∈ L 2 T (W )n such thatX = EX +n∑k=1∫ Talmost surely. Y k is unique λ ⊗ P almost surely.0Y ks dW k sProof. We first show the theorem for variables of the form given in Lemma 3.7.5 andthen extend the result by a density argument.Step 1: A special case. Let h 1 , . . . , h n ∈ L 2 [0, T ] be given, and putX s = exp( n∑k=1∫ s0h k (t) dW k t − 1 2n∑k=1∫ s0h 2 k(t) dtfor s ≤ T . Our goal is to prove the theorem for X T . Note that defining M by puttingM t = ∑ n∫ tk=1 0 h k(s) dWs k , X t = E(M) t and so, by Lemma 3.7.1, X T = 1+ ∫ T0 X s dM s .According to Lemma 3.7.3, X is a square-integrable martingale, and therefore we mayin particular conclude EX T = 1, showingX T = EX T +n∑k=1∫ T0X s h k (s) dW k s .It remains to argue that s ↦→ X s h k (s) is in L 2 (W ). Since X is a square-integrablemartingale, its squared expectation is increasing. We therefore obtainE∫ T0(X s h k (s)) 2 ds =∫ T0h k (s) 2 EX 2 s ds ≤ EX 2 T∫ T0)h k (s) 2 ds,which is finite. Thus, the process s ↦→ X s h k (s) is in L 2 T (W ), and the theorem is provenfor X T .Step 2: The general case. In order to extend the result to the genral case, let Hbe the class of variables in L 2 (F T ) where the theorem holds. Clearly, H is a linearspace. By the first step and Lemma 3.7.5, we have therefore shown the result for adense subspace of L 2 (F T ). The general case will therefore follow if we can show thatH is closed.To do so, assume that X n is a sequence in H converging towards X ∈ L 2 (F T ). ThenX n = EX n + ∑ nk=1∫ T0 Y k n (s) dW k s almost surely, for some processes Y n ∈ L 2 T (W )n .

98 The Itô IntegralSince X n converges in L 2 , the norms and therefore the second moments also converge.Therefore, using Itô’s isometry, Lemma 3.3.6,(n∑‖Yn k − Ym‖ k 2 W = E ∑ n Tk=1k=1∫ T0Y k n (s) − Y k m(s) dW k s= E (X n − EX n − (X m − EX m )) 2= ‖(X n − X m ) − (EX n − EX m )‖ 2 2.Since both ‖X n −X m ‖ 2 and |EX n −EX m | tends to zero as n and m tends to infinity, weconclude that for each k ≤ n, (Yn k ) is a cauchy sequence in L 2 T (W ). By completeness,there exists Y k such that ‖Yn k − Y k ‖ W T tends to zero, and we then obtain) 2X = lim X m mn∑= lim EX m + Y kmm(s) dWsk= EX +k=1n∑Y k (s) dWskk=1almost surely, by the continuity of the integral, where the limits are in L 2 .existence part of the theorem is now proven.TheStep 3: Uniqueness.To show uniqueness, assume that we have two representationsX = EX +The Itô isometry yieldsn∑k=1∫ T0Y ks dW k s = EX +n∑k=1∫ T0Z k s dW k s .n∑‖Y k − Z k ‖ 2 W Tk=1= E ( n∑∫ Tk=10Y k (s) − Z k (s) dW k s) 2= 0,so Y k and Z k are equal λ ⊗ P almost surely, as desired.Theorem 3.7.7 (The Martingale Representation Theorem). Assume that Mis a F t cadlag local martingale. Then, there exists Y ∈ L 2 (W ) n such thatM t = M 0 +n∑k=1∫ t0Y ks dW k s ,almost surely. If M is square-integrable, Y can be taken to be in L 2 (W ) n . Y k is uniqueλ ⊗ P almost surely.

3.7 Itô’s Representation Theorem 99Proof. Uniqueness follows immediately from Lemma 3.3.7. We therefore need onlyconsider existence. Obviously, it will suffice to consider the case where M 0 = 0. Thisis proven in several steps:1. Square-integrable martingales.2. Continuous martingales.3. Cadlag uniformly integrable martingales.4. Cadlag local martingales.Step 1: The square-integrable case. Assume that M is square-integrable startingat zero. In particular, EM = 0. For m ≥ 1, let Y m ∈ L 2 m(W ) n be the process thatexists by Theorem 3.7.6 such thatM m =n∑k=1∫ m0Y k m(s) dW k salmost surely. We then have, by the martingale property of the stochastic integral,n∑∫ (m∑ n ∫ )m+1Ym+1(s) k dWs k = EYm+1(s) k dWsk ∣ F mk=10k=1= E (M m+1 |F m )= M mn∑=k=1∫ m00Y k m(s) dW k s ,almost surely. Then, by the uniqueness part of Theorem 3.7.6, Y k m+1 and Y k m must beλ⊗P almost surely equal on [0, m]×Ω. By the Pasting Lemma 2.5.10, there exists Y ksuch that Y k is almost surely equal to Y k m on [0, m]×Ω. In particular, Y [0, t] ∈ L 2 (W ) nfor all t ≥ 0. This implies that for t ≥ 0,M t = E(M [t]+1 |F t )(∑ n= E= E=∫ [t]+1k=10∫ [t]+1( n∑n∑k=1k=1∫ t00Y ks dW k s ,Y k[t]+1 (s) dW k sY ks dW k s)∣ F t)∣ F t

100 The Itô Integralalmost surely, as desired. By Lemma 3.3.16, Y ∈ L 2 (W ) n .Step 2: The continuous case. Now let M be a continuous martingale startingat zero. Then M is locally a square-integrable martingale. Let τ n be a localisingsequence. By what we already have shown, there exists Y m ∈ L 2 (W ) n such thatMtτm= ∑ n∫ tk=1 0 Y m(s) k dWs k almost surely. We then obtainn∑∫ (t∑ n ∫ ) τmtYm+1[0, k τ m ](s) dWs k = Ym+1(s) k dWskk=10k=10= (M τm+1 ) τ mt= M τ mtn∑=k=1∫ t0Y k m(s) dW k s .Since both integrands are in L 2 t (W ) n , we conclude Y k m+1[0, τ m ] = Y k m λ ⊗ P almostsurely on [0, t]. Letting t tend to infinity, we may then conclude Y k m+1[0, τ m ] = Y k mλ ⊗ P almost surely on all of [0, ∞). By the Pasting Lemma, there exists Y k suchthat Y k [0, τ m ] = Y k m[0, τ m ], λ ⊗ P almost surely. In particular, Y k [0, τ m ] ∈ L 2 (W ), soY ∈ L 2 (W ) n , andM τ mt ===n∑∫ tk=10∫ tn∑k=1( n∑k=10∫ tY k m[0, τ m ] s dW k sY k [0, τ m ] s dW k s0Y ks dW k s) τm,almost surely. Letting m tend to infinity, we get M t = ∑ n∫ tk=1 0 Y s k dWsk almost surely.This shows the theorem in the case where M is a continuous martingale.Step 3: The cadlag uniformly integrable case. Next, let M be a cadlaguniformly integrable martingale starting at zero. By Theorem 2.4.3, M is convergentalmost surely and is closed by its limit, M ∞ . Let M∞ m be a sequence of boundedvariables converging towards M ∞ in L 1 . Assume in particular that ‖M ∞ −M∞‖ m 1 ≤ 13. nM∞ m is square-integrable, so putting Mtm = E(M∞|F m t ), M m is a square-integrablemartingale. By what we already have proven, there are Y m ∈ L 2 (W ) n such thatMtm = M 0 + ∑ n∫ tk=1 0 Y m(s) k dWs k almost surely. In particular, there exists a versionN m of M m in cM 2 0.

3.7 Itô’s Representation Theorem 101Next note that by Doob’s maximal inequality, Theorem 2.4.2,P((N m − M) ∗ t > 12 m )≤ 2 m E|N m t − M t | = 2 m E|M m t − M t |Letting t tend to infinity, we obtain by L 1 -convergenceP((N m − M) ∗ > 12 m )≤ 2 m E|M m ∞ − M ∞ | ≤ 2m3 m .By the Borel-Cantelli Lemma, then, (N m − M) ∗ tends to zero almost surely, so N mconverges almost surely uniformly to M. In particular, M must have a continuousversion, so the conclusion in this case follows from what we proved in the previousstep.Step 4: The cadlag local martingale case. Finally, assume that M is a cadlaglocal martingale starting at zero. By Lemma 2.5.16, M is locally a cadlag uniformlyintegrable martingale. Letting τ m be a localising sequence, we know that the theoremholds for M τm . We can then use the pasting technique from step 2 to obtain thedesired representation for M.The martingale representation theorem has the following important corollary.Corollary 3.7.8. Let M be a F t local martingale. Then there is Y ∈ L 2 (W ) n suchthat M and Y ·W are versions. In particular, any F t local martingale has a continuousversion.Proof. By Theorem 2.4.1, there is a cadlag version of M. We can then apply Theorem3.7.7 to obtain the desired result.Corollary 3.7.8 essentially characterises all martingales with respect to the augmentedBrownian filtration. It provides considerable insight into the structure of F t martingales.In particular, it shows that when we are considering the filtration F t , ourdefinition of the stochastic integral encompasses all local martingales as integrators.Therefore, in the following, whenever we are considering the augmented Brownian filtration,we can assume whenever convenient that we are dealing with continuous localmartingales instead of the class cM L W of stochastic integrals with respect to Brownianmotion.

102 The Itô Integral3.8 Girsanov’s TheoremIn this section, we will prove Girsanov’s Theorem. As in the previous section, welet (Ω, F, P, F t ) be a probability space with a n-dimensional Brownian motion W ,where (Ω, F, P ) is complete and F t is usual augmentation of the filtration induced bythe Brownian motion. We let F ∞ = σ(∪ t≥0 F t ). Girsanov’s Theorem characterisesthe measures which are equivalent to P on F ∞ and shows how the distribution of Wvaries when changing to an equivalent measure. We will see that changing the measurecorresponds to changing the drift of the Brownian motion.The theorem thus has two parts, describing:1. The form of the measures equivalent to P on F ∞ .2. The distribution of W under Q, when Q is equivalent to P on F ∞ .After proving both the results, we will prove Kazamakis and Novikovs conditions onthe existence of measure changes corresponding to specific changes of drift.The first result is relatively straightforward and only requires a small lemma.Lemma 3.8.1. Let Q be a measure on F such that Q is absolutely continuous withrespect to P on F ∞ . Let Q t denote the restriction of Q to F t , and let P t denote therestriction of P to F t . Define the likelihood process L t by L t = dQtdP t. Then L is auniformly integrable martingale, and it is closed by dQ∞dP ∞.Proof. All the claims of the lemma will follow if we can show E( dQ∞dP ∞|F t ) = L t for anyt ≥ 0. To do so, let t ≥ 0 and let A ∈ F t . Using Lemma B.6.8 twice yields∫AdQ ∞dP ∞dP =AA∫AdQ ∞dP ∞dP ∞ =∫A∫dQ ∞ =AdQ t .Making the same calculations in the reverse direction then yields∫ ∫∫∫dQ t dQ tdQ t = dP t = dP = L t dP.dP t dP tThus, E(1 AdQ ∞dP ∞) = E(1 A L t ) for any A ∈ F t , and therefore E( dQ ∞dP ∞|F t ) = L t . Thisshows that L is a martingale closed by dQ ∞dP ∞. In particular, L is uniformly integrableand has the limit dQ ∞dP ∞almost surely and in L 1 .AA

3.8 Girsanov’s Theorem 103Theorem 3.8.2 (Girsanov’s Theorem, part one). Let Q be a measure on Fequivalent to P on F ∞ . There exists a process M ∈ cM L W such that E(M) is auniformly integrable martingale with the property dQ∞dP ∞= E(M) ∞ .Proof. Let L t be the likelihood process for dQdP. By Lemma 3.8.1, L is a uniformlyintegrable martingale. Since the filtration is assumed to satisfy the usual conditions,there exists a cadlag version of L. Since L t is almost surely nonnegative for anyt ≥ 0, we can take the cadlag version to be nonnegative almost surely. By Theorem3.7.7, there exists processes Y ∈ L 2 (W ) n such that we have L t = ∑ n∫ tk=1 0 Y s dW s . Inparticular, L can be taken to be continuous.Our plan is to show that L satisfies an exponential SDE, this will yield the desiredresult. If L almost surely has positive paths, we can do this simply by writingL t =n∑k=1∫ t0YskL s dWs k ,L sand if Y ksL s∈ L 2 (W ), we have the result. Thus, our proof runs in three parts. First,we show that L almost surely has positive paths. Then we argue that Y kL ∈ L2 (W ).Finally, we prove the theorem using these facts.Step 1: L has positive paths almost surely. To show that L almost surely haspositive paths, define τ = inf{t ≥ 0|L t = 0}. Let t ≥ 0. Then (τ ≤ t) ∈ F t . Usingthat dQ tdP t= L t , we obtain∫∫Q(τ ≤ t) = Q t (τ ≤ t) = L t dP t = L t dP.(τ≤t)(τ≤t)Next using that (τ ≤ t) ∈ F τ∧t and that L is a P -martingale, we find∫∫∫∫L t dP = E(L t |F τ∧t ) dP = L τ∧t dP = L τ dP = 0,(τ≤t)(τ≤t)(τ≤t)(τ≤t)where we have used that whenever τ is finite, L τ = 0. This shows Q(τ ≤ t) = 0. SinceP and Q are equivalent, we conclude P (τ ≤ t) = 0. Letting t tend to infinity, weobtain P (τ < ∞) = 0. Therefore, P almost all paths of L are positive. By changingL on a set of measure zero, we can assume that all the paths of L are positive.Step 2: Integrability. Next, we show that Y ksL s∈ L 2 (W ). Let τ n be a localisingsequence for Y k . Define the stopping time σ n by σ n = inf{t ≥ 0|L t < 1 n}. Since the

104 The Itô Integralpaths of L are positive, σ n tends to infinity. By L σ nalmost surely, so ( Y ksM t = ∑ n∫ tk=1 0desired.Y sL sdW k s(∣ ∣∣∣ Y ksL s∣ ∣∣∣) τn ∧σ n≤ n(Y ks ) τ n∧σ n,≥ 1 n, we obtainL s) τ n∧σ nis in L 2 (W ), showing Y ksL s∈ L 2 (W ). Therefore, the processis well-defined, and L t = ∑ n∫ tk=1 0 L s YsL sdWsk = ∫ t0 L s dM s , asStep 3: Conclusion. Finally, by Lemma 3.7.1, we have L = E(M). Since L isuniformly integrable by Lemma 3.8.1 with limit dQ ∞dP ∞, we conclude dQ ∞dP ∞= E(M) ∞ .We have now proved the first part of Girsanov’s Theorem. Theorem 3.8.2 yields aconcrete form for any measure Q which is equivalent to P on F ∞ . The other half ofGirsanov’s Theorem uses this concrete form to describe the Q-distribution of W .Theorem 3.8.3 (Girsanov’s Theorem, part two). Let Q be a measure on Fequivalent to P on F ∞ with Radon-Nikodym derivative dQ ∞dP ∞= E(M) ∞ , where M isin cM L W such that E(M) is a uniformly integrable martingale. Let M = Y · W , whereY ∈ L 2 (W ) n . The process X in R n given by Xtk = Wt k − ∫ t0 Y s k ds is an n-dimensionalF t -Brownian motion under Q.Proof. We first argue that the conclusion is well-defined. Let Y ∈ L 2 (W ) n , and letτ m be a localising sequence such that Y [0, τ m ] ∈ L 2 (W ) n . Then ∫ t0 (Y k [0, τ m ] s ) 2 dsis almost surely finite, and by Lemma B.2.4, ∫ t0 |Y k [0, τ m ] s | ds is almost surely finite.Thus, Y k ∈ L 1 (t), and ∫ t0 Y ks ds is well-defined. Therefore, X k is well-defined and theconclusion is well-defined. Also note that since Y k is progressive, X k is adapted toF t . Therefore, to prove the theorem, it will suffice to show that s ↦→ X t+s − X t is aBrownian motion for any t ≥ 0.To prove the result, we first consider the case where Y is bounded and zero from adeterministic point onwards. Afterwards, we will obtain the general case by approximationand localisation arguments.Step 1: The dominated case, zero from a point onwards. First assume thatthere exists T > 0 such that Y t is zero for t ≤ T and assume that Y is bounded. Havingthis condition will allow us to apply Lemma 3.7.3 in our calculations. To show that

3.8 Girsanov’s Theorem 105X is an F t -Brownian motion under Q, it will suffice to show that for any 0 ≤ s ≤ t,( n)∣ ) (∑∣∣∣∣E(expQ θ k (Xt k − Xs k 1) F s = exp2k=1)n∑θk(t 2 − s) .k=1In order to show this, let f k ∈ L 2 [0, ∞) and A ∈ F. Put Z k u = Y k u + f k (u) andN = Z·W , by Lemma 3.7.3 that the exponential martingale E(N) is a square-integrablemartingale. In particular E(N) has an almost sure limit. We first proveE Q 1 A exp( n∑k=1∫ ∞0f k (u) dX k u)= exp(12n∑k=1∫ ∞0f k (u) 2 du)E P 1 A E(N) ∞ ,and use this formula to show the conditional expectation formula above. Note thatthe stochastic integral above is well-defined in the sense that X k ∈ S under P . Weconsider the left-hand side of the above. Note that from the condition on Y , we obtainY ∈ L 2 (W ) n and therefore ∫ ∞Y k 0 u dWu k and ∫ ∞0 (Y u k ) 2 du are well-defined. We cantherefore writeE Q 1 A exp= E P 1 A exp= E P 1 A exp( n∑∫ ∞k=10∫ ∞( n∑k=10∫ ∞( n∑k=10f k (u) dX k uf k (u) dX k u))f k (u) dX k u +E(M) ∞n∑k=1∫ ∞0Y k u dW k u − 1 2n∑k=1∫ ∞0(Y k u ) 2 du).We note that==∫ ∞0∫ ∞f k (u) dX k +f k (u) dW k u −∫ ∞0∫ ∞00∫ ∞0f k (u) + Y k u dW k u + 1 2Y k u dW k u − 1 2f k (u)Y k u du +∫ t(Y k u ) 2 du0∫ ∞0∫ ∞0f k (u) 2 du − 1 2Y k u dW k u − 1 2∫ ∞0∫ t0(Y k u ) 2 du(f k (u) + Y k u ) 2 du.

106 The Itô IntegralWe then obtainE P 1 A exp= E P 1 A exp= E P 1 A exp= exp(12n∑k=1( n∑∫ ∞k=10∫ ∞( n∑k=1(12n∑k=1∫ ∞00∫ ∞0f k (u) dX k u +n∑k=1∫ ∞f k (u) + Yu k dWu k + 1 2)f k (u) 2 duf k (u) 2 du)0E(N) ∞E P 1 A E(N) ∞ .Y k u dW k u − 1 2∫ ∞0n∑k=1f k (u) 2 du − 1 2We have now proved that for any f k ∈ L 2 [0, ∞) and A ∈ F,E Q 1 A exp( n∑k=1∫ ∞0f k (u) dX k u)= exp(12n∑k=1∫ ∞0f k (u) 2 du∫ ∞0∫ ∞)0(Y k u ) 2 du)(f k (u) + Y k u ) 2 duE P 1 A E(N) ∞ .We are ready to demonstrate that X is an F t -Brownian motion under Q. Let 0 ≤ s ≤ t,A ∈ F s and θ ∈ R n . We can then define f k by f k (u) = θ k 1 (s,t] (u). Since f k is zero on[0, s], M s = N s and therefore, using that E(N) is a uniformly integrable martingale,E P 1 A E(N) ∞ = E P 1 A E(N) s = E P 1 A E(M) s = Q(A).)We then obtain( n)∑E Q 1 A exp θ k (Xt k − Xs k )k=1= E Q 1 A exp= exp(12(1n∑( n∑∫ ∞k=10∫ ∞k=10∫ ∞f k (u) dX k uf k (u) 2 ds)n∑= expf k (u) 2 ds Q(A)2k=10()1n∑= exp θ 22k(t − s) Q(A).k=1))E P 1 A E(N) ∞This shows the desired formula,( n)∣ ) (∑∣∣∣∣E(expQ θ k (Xt k − Xs k 1) F s = exp2k=1)n∑θk(t 2 − s) .Finally, we may conclude that X is an F t -Brownian motion under Q.k=1

3.8 Girsanov’s Theorem 107Step 2: The L 2 (W ) case, deterministic from a point onwards. Now onlyassume that Y ∈ L 2 (W ) n and that Y is zero from T and onwards. In the previousstep, we obtained our results by calculating the conditional Laplace transform. In thecurrent case, we will instead calculate the conditional characteristic function. Our aimis to show that for any 0 ≤ s ≤ t,()∣ ) ()n∑∣∣∣∣E(expQ i θ k (Xt k − Xs k ) F s = exp − 1 n∑θ 22k(t − s) .k=1We will use an approximation argument. To this end, define Yn k = Y k ∧ n ∨ −n.Y n then satisfies the condition of the first step, and we may conclude that puttingM n (t) = Y n · W and defining Q n by dQ ndP n= E(M n ) ∞ , Q n is a probability measure andX n given by Xn(t) k = Wtk − ∫ t0 Y n k (s) ds is a Brownian motion under Q n .Our plan is to show thatk=11. There is a subsequence of E(M m ) ∞ converging to E(M) ∞ in L 1 .2. For t ≥ 0, there is a subsequence of X k m(t) converging to X k (t) almost surely.We will be able to combine these facts to obtain our desired result.To prove the first of the two convergence results above, note that Ynk tends to Y k inL 2 (W ) by dominated convergence. Therefore, ∫ ∞Y k 0 n (s) dWs k tends to ∫ ∞Y k (s) dW k 0 sin L 2 . By taking out a subsequence, we can assume that the convergence is almostsure. Furthermore, from Lemma B.6.5 we find that by taking a subsueqence, we canassume that ∫ ∞0Y k n (s) 2 ds tends to ∫ ∞0Y k (s) 2 ds almost surely. We have now ensuredthat almost surely, ∫ ∞0Y k n (s) dW k s tends to ∫ ∞0Y k (s) dW k s , and ∫ ∞0Y k n (s) 2 ds tendsto ∫ ∞0Y k (s) 2 ds. Therefore, E(M n ) ∞ converges almost surely to E(M) ∞ . We knowthat E P E(M n ) ∞ = 1, and have assumed E P E(M) ∞ = 1. By nonnegativity, Scheffé’sLemma B.2.3 yields that E(M n ) ∞ converges in L 1 to E(M) ∞ , as desired.To prove the second convergence result, let t ≥ 0 be given. Note that since Ynk convergesin L 2 (W ) to Y k and the processes are zero from a deterministic point onwards,we obtain from Lemma B.2.4 that E ∫ ∞|Y k 0 n (s)−Y k (s)| ds tends to zero. Therefore, byLemma B.6.5, we can by taking out a subsequence assume that ∫ t0 Y n k (s) ds convergesalmost surely to ∫ t0 Y k (s) ds. We then obtainlimnX k n(t) = limnW k (t) −∫ t0Y k n (s) ds = X k (t)

108 The Itô Integralas desired.Now let 0 ≤ s ≤ t. We select a subsequence such that1. E(M m ) ∞ converges to E(M) ∞ in L 1 .2. X k m(s) converges almost surely to X k (s).3. X k m(t) converges almost surely to X k (t).Since X m is F t -Brownian motion under Q m , we know that()∣ ) (n∑∣∣∣∣E(expQm i θ k (Xm(t) k − Xm(s))k F s = exp − 1 2k=1We will identify the limit of the left-hand side. We know that( ()∣ )n∑∣∣∣∣E Q mexp i θ k (Xm(t) k − Xm(s))k F s= E P (exp(ik=1)n∑θk(t 2 − s) .k=1) ∣ )n∑∣∣∣∣θ k (Xm(t) k − Xm(s))k E(M m ) ∞ F s .k=1Using our convergence results with Lemma B.2.2, we find that()()n∑n∑exp i θ k (Xm(t) k − Xm(s))k LE(M m ) 1∞ −→ exp i θ k (X k (t) − X k (s)) E(M) ∞ ,k=1and therefore, by the L 1 continuity of conditional expectations,()∣ ) ()n∑∣∣∣∣E(expQ i θ k (Xt k − Xs k ) F s = exp − 1 n∑θ 22k(t − s) ,as desired.k=1k=1k=1Step 3: The L 2 (W ) case. Finally, merely assume Y ∈ L 2 (W ) n . Let σ n be a localisingsequence such that Y k [0, σ n ] ∈ L 2 (W ). Put τ n = σ n ∧ n and define Y k n = Y k [0, τ n ].Then Y k is in L 2 (W ) and is zero from a deterministic point onwards. Therefore,we can use the result from the previous step on Y n . As in the preceeding step ofthe proof, we define M m = Y m · W and define Q m by dQ mdP m= E(M m ) ∞ . PuttingX k m(t) = W k (t) − ∫ t0 Y k m(s) ds, X m is then a F t Brownian motion under Q m . We willshow

3.8 Girsanov’s Theorem 1091. E(M m ) ∞ converges to E(M) ∞ in L 1 .2. X k m(t) converges to X k (t) almost surely.Having these two facts, the conclusion will follow by precisely the same method as inthe previous step. To prove the first convergence result, note that E(M) τ m= E(M m ).Since E(M) is assumed to be a uniformly integrable martingale, it has an almost surelylimit and we may conclude E(M) τm = E(M m ) ∞ . Now, (E(M) τm ) m≥1 is a discretetimemartingale, and it is closed by E(M) ∞ . Therefore, it is convergent in L 1 . ThusLE(M m ) 1∞ −→ E(M) ∞ .The second result follows directly fromlimmX k m(t) = limmW k t −∫ t0Y k m(s) ds∫ t= Wt k − lim Y k (s)1 [0,τm m](s) ds= W k t −= X k t ,∫ t00Y k (s) dsalmost surely. As in the previous step, we conclude that X is an F t Brownian motionunder Q.We are now done with the proof of Girsanov’s Theorem. In the first part of thetheorem, Theorem 3.8.2, we have seen that any probability measure Q equivalent to Pon F ∞ has the property that dQ ∞dP ∞= E(M) ∞ for some M such that E(M) is uniformlyintegrable. In the second part, Theorem 3.8.3, we have seen now the distribution ofW changes under this measure change.When we have two equivalent probability measures P and Q, we will usually be interestedin stochastic integration under both of these probability measures, in the sensethat a given process can be a standard process under P with respect to the originalBrownian motion, and it can be a standard process under Q with respect to the newQ Brownian motion. We have defined a stochastic integral for each of these cases. Wewill now check that whenever applicable, the two stochastic integrals agree. We alsoshow that being a standard process under P is the same as being a standard processunder Q.

110 The Itô IntegralTheorem 3.8.4. Let P and Q be equivalent on F ∞ . If X is a standard process underP , it is a standard process under Q as well. Let L P (X) denote the integrands for Xunder P , and let L Q (X) denote the integrands for X under Q. If Y ∈ L P (X)∩L Q (X),then the integrals under P and Q agree up to indistinguishability.Comment 3.8.5 The meaning of the lemma is a bit more convoluted that it mightseem at first. To be a standard process, it is necessary to work in the context of afiltered space (Ω, F, P, F t ) with a F t Brownian motion. That X is a standard processunder Q thus assumes that we have some particular process in mind which is a F tBrownian motion under Q. In our case, the implicitly understood Brownian motion isthe one given by Girsanov’s theorem.◦Proof. We need to prove two things: That X is a standard process under Q, and thatif ∈ L P (X) ∩ L Q (X), the integrals agree.Step 1: X is a standard process under Q. Let the canonical decomposition ofX under P be given by X = A + M with M = Y · W . By Girsanov’s Theorem, thereis a continuous local martingale N = Z · W such that E(N) is a uniformly integrablemartingale and dQ∞dP ∞= E(N) ∞ , and then the process W Q given byW Q k (t) = W k(t) −∫ t0Z k (s) dsis a F t Brownian motion under Q. Note that since the null sets of P and Q arethe same, the space (Ω, F, Q, F t ) satisfies the usual conditions. Thus, with W Q as ourBrownian motion, we have a setup precisely as described throughout the chapter. Andby the properties of the stochastic integral under P ,X t = A t + M tn∑= A t += A t +∫ tk=10∫ tn∑k=10Y k (s) dW k (s)Z k (s) ds +n∑k=1∫ t0Y ks dW Q k (s).Under the setup (Ω, F, Q, F t ), the process A t + ∑ nk=1 Z k(s) ds is of finite variation andthe process ∑ nk=1 Y s k dW Q k (s) is in cML W . Therefore, X is a standard process underQ.Step 2: Agreement if the integrals, part 1. Now assume that Y is some processwith is in L P (X) ∩ L Q (X). Let IX P (Y ) be the integral of Y with respect to X under

3.8 Girsanov’s Theorem 111P , and let I Q X(Y ) be the integral of Y with respect to X under Q. We want to showthat IX P (Y ) and IQ X(Y ) are indistinguishable.We first consider the case where Y is bounded and continuous. Let t ≥ 0. We willshow that IX P (Y ) t = I Q X (Y ) t almost surely. By Lemma 3.4.15, the Riemann sums∑ nk=1 Y t k−1(X tk − X tk−1 ) converge in probability under P to IX P (Y ) t and converge inprobability under Q to I Q X (Y ) t for some sequence of partitions of [0, t] with meshestending to zero. By taking out a subsequence, we can assume that the convergenceis almost sure. Since P and Q are equivalent, we find that both under P and Q, theRiemann sums tend to both IX P (Y ) t and I Q X (Y ) t. Therefore, IX P (Y ) t and I Q X (Y ) t arealmost surely equal. Since both processes are continuous, we conclude that IX P (Y ) andI Q X(Y ) are indistinguishable.Step 3: Agreement of the integrals, part 2. Now, as before, let X = A + Mbe the canonical decomposition of X under P , and let X = B + N be the canonicaldecomposition of X under Q. We consider the case where V A and V B are bounded,where M, N ∈ cM 2 W and where Y is bounded and zero from a deterministic pointonwards. As in the proof of Theorem 3.1.7, defineY n (t) = 1 1n∫ t(t− 1 n )+ Y (s) ds.Then Y n is bounded and continuous with ‖Y n ‖ ∞ ≤ ‖Y ‖ ∞ , and Y n converges to Yλ ⊗ P everywhere. Since µ M and µ N are bounded, this implies that Y n convergesto Y in L 2 (M) and L 2 (N), and therefore IM P (Y n) converges to IM P (Y ) in cM2 0 underP . Likewise, I Q N (Y n) converges to I Q N(Y ) under Q. Taking out subsequences, wecan assume that the convergence is uniformly almost sure, showing in particular thatIM P (Y n) t converges to IM P (Y ) t almost surely and I Q N (Y n) t converges to I Q N (Y ) t almostsurely for any t ≥ 0.Next, consider the finite variation components. Since V A and V B are bounded we findthat for almost all t ≥ 0, I P A (Y n) t converges almost surely to I P A (Y ) t. Likewise, foralmost all t ≥ 0, I Q B (Y n) t converges almost surely to I Q B (Y ) t.All in all, we may conclude that for almost all t ≥ 0, I P X (Y n) t converges to I P X (Y ) tand I Q X (Y n) t converges to I Q X (Y ) t almost surely. Therefore, by what we already haveshownI P X(Y ) t = limnI P X(Y n ) t = limnI Q X (Y n) t = I Q X (Y ) t.Since any Lebesgue almost sure set is dense and the processes are continuous, we

112 The Itô Integralconclude that IX P (Y ) and IQ X(Y ) are indistinguishable.Step 4: Agreement of the integrals, part 3. We now still assume that V A andV B are bounded and that M, N ∈ cM 2 W , but we reduce our requirement on Y to bethat Y is in L 1 (A) ∩ L 1 (B) ∩ L 2 (M) ∩ L 2 (N). We will use the results of the previousstep to show that the result holds in this case. DefineY n (t) = Y (t)1 (|Yt |>n)1 [0,n] (t).Then Y is bounded and zero from a deterministic point onwards. Furthermore,|Y n (t)| ≤ |Y (t)|, and Y n converges pointwise to Y . By dominated convergence, wetherefore conclude that Y n converges to Y in L 2 (M) and L 2 (N). This implies thatIM P (Y n) tends to IM P (Y ) in cM2 0 under P , and I Q N (Y n) tends to I Q N (Y ) in cM2 0 underQ. For the finite variation compoments, we find that IA P (Y n) converges pointwise toIA P (Y ) and IQ A (Y n) converges pointwise to I Q A(Y ).As in the previous step, this impliesshowing the result in this case.I P X(Y ) t = limnI P X(Y n ) t = limnI Q X (Y n) t = I Q X (Y ) t,Step 5: Agreement of the integrals, part 4. Finally, we consider the generalcase. Define the stopping timesτ A n = inf{t ≥ 0|V A (t) ≥ n}τ B n = inf{t ≥ 0|V A (t) ≥ n},and let τn M and τn N be sequences localising M and N to cM 2 W , respectively. Let σ nbe such that Y σn is in L 1 (A) ∩ L 1 (B) ∩ L 2 (M) ∩ L 2 (N). Defining the stopping timeτ n by τ n = τn A ∧ τn B ∧ τn M ∧ τn N ∧ σ n . ThenV τn τnAand VB are bounded, M τn and N τnare in cM 2 W and Y τ nis in L 1 (A τ n) ∩ L 1 (B τ n) ∩ L 2 (M τ n) ∩ L 2 (N τ n). Furthermore, τ ntends to infinity almost surely. By Lemma 3.4.11,I P X(Y ) τn = I P X τ n (Y )I Q X (Y )τn = I Q X τn (Y ).Since P and Q are equivalent, τ n converges to infinity both Q and P almost surely.Therefore, since we have shown the result in the localised case, the result follows forthe general case as well.

3.8 Girsanov’s Theorem 113Next, we consider necessary requirements for the practical application of the Girsanovtheorem. In practice, we are not given a probability measure Q and want to identifythe distribution of W under Q. Rather, we desire to change the distribution of W ina certain manner, and wish to find an equivalent probability measure Q to obtain thischange. If we plan to use Theorem 3.8.3 to facilitate this change, we need to identifya local martingale M zero at zero yielding the desired distribution and ensure thatE(M) is a uniformly integrable martingale. This latter part is the most difficult partof the plan. Note that the condition M 0 = 0 is necessary for E(M) to be defined atall.In Lemma 3.7.3, we obtained a sufficient condition to ensure that E(M) is uniformlyintegrable. This condition is too strong to be useful in practice, however. Our nextobjective is to find weaker conditions that ensure the uniformly integrable martingaleproperty of E(M). These conditions take the form of Kazamaki’s and Novikov’scriteria. We are led by Protter (2005), Section III.8.Lemma 3.8.6. Let M ∈ cM L 0 . E(M) is a supermartingale with EE(M) t ≤ 1 for allt ≥ 0. E(M) is a martingale if and only if EE(M) t = 1 for all t ≥ 0. E(M) is auniformly integrable martingale if and only if EE(M) ∞ = 1.Proof. Since E(M) is a continuous nonnegative local martingale, it is a supermartingale.In particular, EE(M) t ≤ EE(M) 0 = 1. Furthermore, E(M) ∞ always exists asan almost sure limit by the Martingale Convergence Theorem.The martingale criterion. We now prove that E(M) is a martingale if and only ifEE(M) t = 1 for t ≥ 0. Clearly, if E(M) is a martingale, EE(M) t = EE(M) 0 = 1 forall t ≥ 0. Conversely, assume that EE(M) t = 1 for all t ≥ 0. Consider 0 ≤ s ≤ t. Bythe supermartingale property, we obtain E(M) s ≥ E(E(M) t |F s ). This shows that0 ≤ E(E(M) s − E(E(M) t |F s )) = EE(M) s − EE(M) t = 0,so E(M) s − E(E(M) t |F s ) is a nonnegative variable with zero mean. Thus, it is zeroalmost surely and E(M) s = E(E(M) t |F s ), showing the martingale property.The UI martingale criterion. Next, we prove that E(M) is a uniformly integrablemartingale if and only if EE(M) ∞ = 1. If EE(M) ∞ = 1, we obtain1 = EE(M) ∞ = E lim inftE(M) t ≤ lim inf EE(M) t .tNow, EE(M) t is decreasing and bounded by one, so we conclude lim t EE(M) t = 1 and

114 The Itô Integraltherefore EE(M) t = 1 for all t. Therefore, by what we already have shown, E(M) is amartingale. By Lemma B.5.4, it is uniformly integrable.On the other hand, if E(M) is a uniformly integrable martingale, it is convergent inL 1 and in particular EE(M) ∞ = lim t EE(M) t = 1.Lemma 3.8.7. Let M be a continuous local martingale zero at zero. Let ε > 0. Ifit holds that sup τ E exp (( 12 + ε) M τ)< ∞, where the supremum is over all boundedstopping times, then E(M) is a uniformly integrable martingale.Proof. Let ε > 0 be given. Our plan is to identify an a > 1 such that sup τ EE(M) a τis finite, where the supremum is over all bounded stopping times. If we can do this,Lemma 2.5.19 shows that E(M) is a martingale bounded in L a , in particular E(M) isa uniformly integrable martingale. To this end, let a, r > 1 be given. We then have(E(M) a τ = exp aM τ − a ) (√ a2 [M] τ = expr M τ − a ) (( √ ) ) a2 [M] τ exp a − M τ .rThe point of this observation is that when we raise the first factor in the right-handside above to the r’th power, we obtain an exponential martingale. Therefore, lettings be the dual exponent of r, Hölder’s inequality and Lemma 3.8.6 yieldsEE(M) a τ≤( ( √arMτE exp − ar )) 1( (( √ ) ))2 [M] ra 1 sτ E exp a − sM τr= ( EE( √ () 1rarM) τ E exp≤( (( √ ) )) a 1 sE exp a − sM τ .r(( √ ) )) a 1 sa − sM τrRecalling that s =rr−1, we then findsup EE(M) a ττ≤=((sup E expτsup E expτ(( √ ) a ra −rr − 1 M τ( (ar−√ ar) 1r − 1 M τ)) 1 s)) 1s.Thus, if we can identify a, r > 1 such that (ar − √ ar) 1r−1 ≤ 1 2+ε, the result will followfrom our assumption. To this end, define f(y, r) = (y − √ y) 1r−1. Finding a, r > 1 suchthat (ar− √ ar) 1r−1 ≤ 1 2 +ε is equivalent to finding y > r > 1 such that f(y, r) ≤ 1 2 +ε.

3.8 Girsanov’s Theorem 115To this end, note that ddy (y − √ y) = 1 − 12 √ y , which is positive whenever y > 1 4 . Wetherefore obtaininf f(y, r) = inf infr − √ rf(y, r) = infy>r>1 r>1 y>r r>1 r − 1= infr>1√ r√ r + 1= infr>111 + 1 √ r= 1 2 .We may now conclude that for any ε > 0, there is y > r > 1 such that f(y, r) ≤ 1 2 + ε.The proof of the lemma is then complete.Comment 3.8.8 This lemma parallels the lemma preceeding Theorem 44 of SectionIII.8 in Protter (2005). However, Protter avoids the analysis of the mapping f used inthe proof above by formulating his lemma differently and making some very inspired√ pchoices of exponents. He shows that if sup τ E exp(2 √ p−1 M τ ) is finite, then E(M) is amartingale bounded in L q , where q is the dual exponent to p. His method is based onHölder’s inequality just as our proof, and he uses r = √ p+1√ p−1.We have opted for a different proof, involving analysis of the mapping f, in order toavoid having to use the very original choice r = √ p+1√ p−1out of the blue.◦Theorem 3.8.9 (Kazamaki’s Criterion). Let M be a continuous local martingalezero at zero. If sup τ E exp ( 12 M τ)< ∞, where the supremum is over all boundedstopping times, then E(M) is a uniformly integrable martingale.Proof. Our plan is to show that E(aM) is a uniformly integrable martingale for any0 < a < 1 and let a tend to one in a suitable context. Let 0 < a < 1. Then there isε a > 0 such that a( 1 2 +ε a) ≤ 1 2. Therefore, with the supremum over bounded stoppingtimes,(( ) )1sup E expτ2 + ε a aM τ≤≤≤((( ) ))1sup E 1 (Mτ 0 such that x + ≤ C + exp( 1 2x). Therefore, by ourassumptions, M t+ is bounded in L 1 , and therefore M t converges almost surely to its

116 The Itô Integrallimit M ∞ . Since the limit [M] ∞ always exists in [0, ∞], we can then write)E(aM) ∞ = exp(aM ∞ − a22 [M] ∞)= exp(a 2 M ∞ − a22 [M] ∞ exp ( (a − a 2 ))M ∞= E(M) a2 exp ( (a − a 2 )M ∞).Using that E(aM) is a uniformly integrable martingale and applying Hölder’s inequalitywith 1 a 2 and its dual exponent11−a 2 , we obtain by Lemma 3.8.6 that1 = EE(aM) ∞≤( a − a2(EE(M))(E a2 exp1 − a 2 M ∞)) 1−a2Now, if the last factor on the right-hand side is bounded over a > 1, we can let a tendto 1 from above and obtain 1 ≤ EE(M) ∞ , which would give us the desired conclusion.Since a−a21−a= a2 1+a = 1 < 1 1a +1 2 , it will suffice to prove that E exp( a 2 M ∞) is boundedover 0 < a < 1. This is what we now set out to do..Define C = sup τ E exp( 1 2 M τ ), C is finite by assumption. Note that since a < 1, thereis p > 1 such that ap ≤ 1 and therefore( ap)2 M t ≤ C.supt≥0( a pE exp t)2 M = sup E expt≥0This shows that exp ( a2 M t)is bounded in L p for t ≥ 0. In particular, the family isuniformly integrable. Since M t converges almost surely to M ∞ , we have convergencein probability as well. Therefore, exp( a 2 M t) converges in L 1 to exp( a 2 M ∞), and wemay conclude( a)( a)E exp2 M ∞ = lim E expt 2 M t ≤ C.In particular, then, we finally get 1 ≤ (EE(M) ∞ ) a2 C 1−a2 , and letting a tend to oneyields 1 ≤ EE(M) ∞ . By Lemma 3.8.6, the proof is complete.Theorem 3.8.10 (Novikov’s Criterion). Let M be a continuous local martingalezero at zero. Assume that E exp( 1 2 [M] ∞) is finite. Then E(M) is a uniformly integrablemartingale.Proof. We will show that Novikov’s criterion implies Kazamaki’s criterion. Let τ beany bounded stopping time. Then( ) ( (E(M) 1 2 1τ = exp2 M τ exp − 1 )) 12 [M] 2τ ,

3.9 Why The Usual Conditions? 117which, using the Cauchy-Schwartz inequality, shows thatE exp( 12 M τ)= EE(M) 1 2 τ(exp( 12 [M] τ≤ (EE(M) τ ) 1 2≤)) 12( ( )) 11E exp2 [M] 2τ( ( )) 11E exp2 [M] 2∞ ,where we have used that [M] is increasing. Since the final right-hand side is independentof τ and finite by assumption, Kazamaki’s criterion is satisfied and the conclusionfollows from Theorem 3.8.9.This concludes our work on the stochastic calculus. In this section, we have provenGirsanov’s theorem, characterising the possible measure changes on F ∞ and givingthe change in drift of the Brownian motion. The Kazamaki and Novikov criteria canbe used to obtain the existence of measure changes corresponding to specific drifts. Ofthese two, the Kazamaki criterion is the strongest, but the Novikov criterion is ofteneasier to apply.Before proceeding to the next chapter, we will in the next section discuss our choiceto assume the usual conditions.3.9 Why The Usual Conditions?The question of whether to assume the usual conditions is a vexed one. Most bookseither do not comment the problem or choose to assume the usual conditions withoutstating why. We have assumed the usual conditions in this text. We will now explainthis choice. We will also describe how we could have avoided having to make thisassumption, and we will discuss general pros and cons of the usual conditions.A review of the usual conditions. We begin by reiterating the meaning of theusual conditions and reviewing some opinions expressed about the usual conditionsin the literature. Recall that for a filtered probability space (Ω, F, P, F t ), the usualconditions are conditions states that:

118 The Itô Integral1. (Ω, F, P ) is a complete measure space.2. The filtration is right-continuous, F t = ∩ s>t F s .3. For t ≥ 0, F t contains all P -null sets in F.The books Rogers & Williams (2000a) and Protter (2005) both have sections discussingthe effects and the reasonableness of assuming the usual conditions.In Rogers & Williams (2000a), Section II.76, the right-continuity of the filtration istaken as a natural choice. The authors focus on why the completion of the filtration isto be introduced. The main argument is that this is necessary for the Début Theoremand Section Theorem of Section II.77 to hold. Of these two, the Début Theorem isthe easiest to understand, simply stating that for any progressive process X and Borelset B, the variable D B = inf{t ≥ 0|X ∈ B} is a stopping time. This is not necessarilytrue if the filtration is not completed. The reason for this is illustrated in the proofof Lemma 75.1, where it is explained how the process can approach B in uncountablymany ways, which causes problems with measurability.In Protter (2005), Section 1.5, the completion of the filtration is instead seen as anatural choice, and the author focuses on why the right-continuity should be included.He does this by showing that the completed filtration of any cadlag Feller process isautomatically right-continuous, so when assuming completion, right-continuity comesnaturally.While enlightening, none of these discussions provide reasons for us to assume theusual filtrations. Our reasons for assuming the usual conditions are much more downto-earthand actually motivated from mostly practical concerns. Before discussingupsides and downsides from assuming the usual conditions in our case, we make somegeneral observations regarding the effects of the usual conditions.Regularity of paths and the usual conditions. We take as starting point thetopic of cadlag versions of martingales. We claim that1. Without any assumptions on the filtration, no cadlag versions are guaranteed.2. With right-continuity, almost sure cadlag paths can be obtained.3. With right-continuity and completeness, cadlag paths for all ω can be obtained.

3.9 Why The Usual Conditions? 119The first claim is basically unfounded, as we have no counterexample to show that thereare martingales where no cadlag versions exists. However, there are no theorems in theliterature to be found giving results on general filtrations, and from the structure ofthe proofs for right-continuous filtrations, it seems highly unlikely that cadlag versionsshould exist for the general case.To understand what is going on in the two other cases, we follow Rogers & Williams(2000a), sections II.62 through II.67. We work in a probability space (Ω, F, P, F t )where we assume that the filtration is right-continuous. Let x : [0, ∞) → R be somemapping. We define lim q↓↓t x q as the limit of x q as q converges downwards to t throughthe rationals. Likewise, we define lim q↑↑t x q as the limit of x q as q converges upwardsto t through the rationals. We say that x is regularisable if lim q↓↓t x q exists for allt ≥ 0 and lim q↑↑t x q exists for all t > 0. As described in Rogers & Williams (2000a),Theorem II.62.13, if x is regularisable, there exists another function y such that y iscadlag and y agrees with x on [0, ∞) ∩ Q. As shown in Sokol (2007), Lemma 2.73,being regularisable is actually both a sufficient and necessary condition for this to hold.y is explicitly given by y t = lim q↓↓t x t .Now let M be a martingale and defineG = {ω ∈ Ω|M(ω) is regularisable}.By Rogers & Williams (2000a), Theorem II.65.1, G is then F-measurable and it holdsthat P (G) = 1. We define{lim q↓↓t M t (ω) if the limit exists.Y t (ω) =0 otherwise.By a modification of the argument in Rogers & Williams (2000a), Theorem II.67.7, ifthe filtration is right-continuous, Y is then a version of M. Note thatG ⊆ (Y t = limq↓↓tM t ∀ t ≥ 0),and therefore Y is almost surely cadlag. Since the filtration is right-continuous, itis clear that Y is adapted, so Y is an F t martingale. Thus, under the assumption ofright-continuity, we have been able to use Rogers & Williams (2000a) Theorem II.67.7to obtain a version Y of M which is a martingale and almost surely cadlag.We will now consider what can be done to make sure that all paths of our version of Mare cadlag. An obvious idea would be to put Y ′ = 1 G Y . This corresponds to letting Y ′be equal to Y on G, where Y is guaranteed to be cadlag, and zero otherwise. All paths

120 The Itô Integralof Y ′ are cadlag. However, since we have no guarantee that G ∈ F t , Y ′ is no longeradapted. This is what ruins our ability to obtain all paths cadlag in the uncompletedcase: the set of non-cadlag paths does not interact well with the filtration.However, if we assume that F t contains all P -null sets, we can use the fact thatP (G) = 1 to see that Y ′ is adapted. Thus, in this case, we can obtain a version Y ′ ofM which is a martingale and has all paths cadlag.We have now argued for the three claims we made earlier. A general moral of ourarguments is that if X is an adapted process, then we can change X on a null set andretain adaptedness if the filtration contains all null sets.Upsides and downsides of the usual conditions. We now discuss reasons forassuming or not assuming the usual conditions.The upsides of the usual conditions are:1. Path properties hold for all ω, so there are fewer null sets to worry about.2. Proofs of adaptedness are occasionally simpler.3. Most literature assumes the usual conditions, making it easier to refer results.The downsides of the usual conditions are:1. We need to spend time checking that, say, the Brownian motion can be taken tobe with respect to a filtration satisfying the usual conditions.2. Density proofs are occasionally harder.To see how the usual conditions can make density proofs harder, look at the proofof Lemma 3.7.5. To see how proofs of adaptedness can be harder without the usualconditions, we go back to the construction seen earlier, Y t (ω) = lim q↓↓t M t (ω) wheneverthe limit exists and zero otherwise. Had we assumed the usual conditions, wecould just define the entire trajectory of Y on some almost sure set and putting it tozero otherwise. This phenomenon would also cause some trouble when defining thestochastic integral for integrators of finite variation.

3.10 Notes 121As null sets really are a frustrating aspect of the entire theory, adding almost no realvalue but constantly complicating proofs, we have chosen to offer a bit of extra effortto introduce the usual conditions, in return obtaining fewer null sets to worry about.We could have developed the entire theory without the usual conditions, but then allour stochastic integrals and other processes would only be continuous almost surely.Some would say that another downside of the usual conditions is that the filtrationloses some of its interpretation as “carrier of information”, in the sense that F t isusually interpreted as “the information available at time t”. We do not find thisargument to be particularly weighty. Obviously, the null sets N contain sets from allof F, in particular sets from F t for any t ≥ 0. But these are sets of probability zero. Itwould therefore seem that the information value in these sets is quite minimal. Sincecompleting the filtration does not have any negative impact on actual modeling issues,it seems unreasonable to maintain that the interpretation of the filtration is reduced.There do seem to be situations, though, where the usual conditions should not beassumed. When considering the martingale problem, Jacod & Shiryaev (1987) inSection III.1 specifically notes that they are considering filtrations that do not necessarilysatisfy the usual conditions. However, Rogers & Williams (2000b) also considersthe martingale problem in Section V.19, even though they have only developedthe stochatic integral under the usual conditions, and seem to get through with this.Changes of measure can also present situations where the usual conditions are inappropriate,see the comments in Rogers & Williams (2000b), Section IV.38, page82.3.10 NotesIn this section, we will review other accounts of the theory in the literature, and wewill give references to extensions of the theory. We will also compare our methods tothose found in other books and discuss the merits and demerits of our methods.Other accounts of the theory of stochastic integration. There are many excellentsources for the theory of stochastic integration. We will consider the approachesto the stochastic integral taken in Øksendal (2005), Rogers & Williams (2000b),Protter (2005) and Karatzas & Shreve (1988).

122 The Itô IntegralIn Øksendal (2005), the integrators are resticted to processes of the type given by∫ t0 Y t dt+ ∫ t0 Y t dW t . The integral is developed over compact intevals and the viewpointis more variable-oriented and less process-oriented. The level of rigor is lower, but thetext is easier to comprehend and gives a good overview of the theory.In Rogers & Williams (2000b), a very systematic development of the integral is made,resulting in a large generality of integrators. As integrators, the authors first takeelements of cM 2 0. They develop the integral for integrands in bE, and then extendto the space of locally bounded predictable integrands, L 0 (bP). Here, the predictableσ-algebra the σ-algebra on (0, ∞) × Ω generated by the left-continuous processes, orequivalently, generated by bE. This σ-algebra is strictly smaller than Σ π , and thusthe integrands of Rogers & Williams (2000b) are less general than ours. However, theextension method is simpler, based on the monotone class theorem. Their greater rangeof integrators comes at the cost of having to develop the quadratic variation in greaterdetail. After constructing the integral for integrators in cM 2 0, they later develop amuch more general theory, where the integrators are semimartingales, meaning thesum of a local martingale and a finite variation process.In Protter (2005), a kind of converse viewpoint to other books is taken. Instead ofdeveloping the integral for more and more general processes, the author defines a semimartingaleas a process X making the integral mapping with respect to X continuousunder suitable topologies. In the first part of the book, the integral is developed asa mapping from left-continuous processes to right-continuous processes. In time, itis shown that this integral leads to the same result as the conventional approach. Inlater parts of the book, the space of integrands are extended to predictable processes.Finally, in Karatzas & Shreve (1988), the integral is developed in the usual straightforwardmanner as in Rogers & Williams (2000b), but here, only continuous integratorsare considered. As in Rogers & Williams (2000b), the authors begin with processes incM 2 0 as integrators, but in this case, the space of integrands is larger than in Rogers& Williams (2000b). The account is very detailed and highly recommended.Extensions of the theory. As noted above, the theory we have presented can beextended considerably. Rogers & Williams (2000b), Protter (2005) and Karatzas& Shreve (1988) all consider more general integrators than we do here. A verypedagogical source for the general theory is Elliott (1982), though the number ofmisprints can be frustrating.

3.10 Notes 123The idea of extending the stochastic integral begs the question of whether resultssuch as Itô’s representation theorem and Girsanov’s theorem also can be extended tomore general situations. To a large extent, the answer is yes. For extensions of therepresentation theorem, see Section III.4 of Jacod & Shiryaev (1987) or Section IV.3of Protter (2005). Extensions of the Girsanov theorem can be found in Section III.3of Jacod & Shiryaev (1987), Section III.8 of Protter (2005), and also in Rogers &Williams (2000b) and Karatzas & Shreve (1988). One immediate extension is theopportunity to prove Girsanov’s theorem in a version where P and Q are not equivalenton F ∞ , but only on F t for all t ≥ 0. Having proved this extension would have madeour applications to mathematical finance easier. Alas, this realization came too late.The search for results like Kazamaki’s or Novikov’s criteria guaranteeing that an exponentialmartingale is a uniformly integrable martingale can reasonably be said tobe an active field of research. See Protter & Shimbo (2006) for recent results andfurther references. For an example of a situation where Kazamaki’s criterion appliesbut Novikov’s criterion does not, see Revuz & Yor (1998), page 336.Discussion of the present account. The account of the theory developed here hasthe same level of ambition as, say, Øksendal (2005) or Steele (2000), but with ahigher level of rigor. It is an attempt to combine the methods of Steele (2000) withthe stringency of Rogers & Williams (2000b) and Karatzas & Shreve (1988).The restriction to continuous integrators simplifies the theory immensely, more or lessto a point beyond comparison. One consequence of this restriction is that we can takeour integrands to be progressively measurable instead of predictably measurable.Not only is progressive measurability less restrictive than predictable measurability,but it also removes some of the trouble with the timepoint zero. Traditionally, therole of zero has not been well-liked. In Dellacherie & Meyer (1975), Section IV.61, theauthors claim that zero plays “the devil’s role”, and in the introduction to ChapterIV of Rogers & Williams (2000b), it is “consigned to hell”. These problems verymuch has something to do with the predictable σ-algebra, this is most obvious inRogers & Williams (2000b) where the predictable σ-algebra is defined as a σ-algebraon (0, ∞) × Ω instead of [0, ∞) × Ω. Even in our case, however, there are someconsiderations to be made. The fundamental problem with zero is that the intuitiverole of an integrator X should only depend on differences X t − X s , interpreted asthe measure assigned to (s, t]. The existence of a value at zero is therefore in factsomething of a nuisance. In our definition of the integral, we have chosen to assign

124 The Itô Integralany non-zero value at zero of the integrator X to the finite-variation part of X andthus deferred the problems to results from ordinary measure theory.In fact, a good deal of our results can be extended to integrators which are not evenprogressively measurable. This is due to a result, the Chung-Doob-Meyer theorem,stating that for any measurable and adapted process, there exists a progressive versionof that process. This would enable us to define the stochastic integral with respect to allmeasurable and adapted processes satisfying appropriate integrability conditions. Theoriginal proof of the Chung-Doob-Meyer theorem was quite difficult and can be foundin Dellacherie & Meyer (1975), Section 30 of Chapter IV, page 99. Recently, a muchsimpler proof has surfaced, see Kaden & Potthoff (2004). Because we have assumedthe usual conditions, we could easily have adapted our results to consider meaurableand adapted integrands. Without the usual conditions, however, the integration ofmeasurable and adapted processes causes problems with adaptedness. In essence,progressive meaurability is the “right” measurability concept for dealing with pathwiseLebesgue integration.Now, as stated in the introduction, part of the purpose of the account given here wasto explore what it would be like to develop the simple theory of stochastic integrationwith respect to Brownian integrators, just as in Steele (2000), but with complete rigorand in a self-contained manner. The reason to do so is mainly to avoid the trouble ofhaving to prove the existence of the quadratic variation process for general continuouslocal martingales. This could potentially allow for a very accesible presentation of thebasic theory of stochastic integration, which covers much of the results necessary for,say, basic mathematical finance. However, as we have seen, the lower level of generalityhas its price. The main demerits are:1. The lack of general martingales makes the theory less elegant.2. The development of the integral through associativity in Section 3.3 is tedious.The need to consider n-dimensional Brownian motion and corresponding n-dimensional integrands makes the notation cumbersome.3. The proof of Girsanov’s theorem of Section 3.8 is very lengthy.The first two demerits more or less speak for themselves. The third demerit deservessome comments. In Øksendal (2005), the integral is also only developed for Brownianintegrators, but the proof of the Girsanov theorem given there is considerably shorter.

3.10 Notes 125This is because the author takes the Lévy characterisation of Brownian motion asgiven, and thus his account is not self-contained.The modern proof of the Lévy characterisation theorem uses a more general theoryof stochastic integration than presented here, see Theorem 33.1 of Rogers & Williams(2000b). There is an older proof, however, not based on stochastic integration theory,but it is much more cumbersome. A one-dimensional proof is given in Doob (1953).If we were to base our proof of the Girsanov theorem on the Lévy characterisationtheorem, our account would become even longer than it is at present. The proofwe have given is based on the proof in Steele (2000) for the case where the driftis bounded. Steele (2000) claims on page 224 that the proof for the bounded caseimmediately can be extended to the general case. It is quite unclear how this shouldbe possible, however, since it would require the result that if E(M) is a martingale andf is bounded and deterministic, then E(M + f · W ) is a martingale as well. Our resultextends the method in Steele (2000) by a localisation argument, and it is to a largedegree this localisation which encumbers the proof.If we had the theory of stochastic integration for general continuous local martingalesat our disposal, we could have used the arguments of Rogers & Williams (2000b),Section IV.33 and Section IV.38, to give short and elegant proofs of both the Lévycharacterisation theorem and the Girsanov theorem.Our conclusion is that when a high level of rigor is required, it is quite doubtful thatthere is any use in developing the theory only for the case of Brownian motion insteadof considering general continuous local martingales. A theory at the level of Karatzas& Shreve (1988) would therefore be preferable as introductory, with works such asØksendal (2005) and Steele (2000) being useful for obtaining a heuristic overview ofthe theory.

126 The Itô Integral

Chapter 4The Malliavin CalculusIn this section, we will define the Malliavin derivative and prove its basic properties.The Malliavin derivative and the theory surrounding it is collectively known as theMalliavin calculus. The Malliavin derivative is an attempt to define a derivative ofa stochastic variable X in the direction of the underlying probability space Ω. Obviously,this makes no sense in the classical sense of differentiability unless Ω has sufficientstructure. The Malliavin derivative is therefore defined for appropriate variables ongeneral probability spaces not by a limiting procedure as in the case of classical derivativeoperators, but instead by analogy with a situation where the underlying space hassufficient topological structure.We will not be able to develop the theory sufficiently to prove the results which are appliedto mathematical finance. Therefore, the exposition given here is mostly designedto give a rigorous introduction to the fundamentals of the theory.Before beginning the development of the Malliavin calculus, we make some motivatingremarks on the definition of the derivative. We are interested in defining the derivativeon probability spaces (Ω, F, P ) with a one-dimensional Wiener process W on [0, T ],where the variables to be differentiated are in some suitable sense related to W . Withthis in mind, we first consider the case where our probability space is C 0 [0, T ], thespace of continuous real functions on [0, T ] starting at zero, endowed with the Wienermeasure P . C 0 [0, T ] is a Banach space under the uniform norm. Let W be the identitymapping, W is then a Wiener process on [0, T ]. We want to identify a differentiability

128 The Malliavin Calculusconcept for variables of the form ω ↦→ W t (ω) which can be generalized to the setting ofan abstract probability space with a Wiener process. To this end, consider a mappingX from C 0 [0, T ] to R. Immediately available differentiability concepts for the mappingare:1. The Frechét derivative: The Frechét derivative of X at ω is the linear operatorA ω : C 0 [0, T ] → R such that lim h→0‖X(ω+h)−X(ω)−A ω (h)‖‖h‖= 0.2. The Gâteaux directional derivatives: The derivative in direction h ∈ C 0 [0, T ] atX(ω+εh)−X(ω)ω is the element D h X(ω) ∈ R such that lim ε→0 ε= D h X(ω).Here, the Frechét derivative can obviously be ruled out as a candidate for generalizationbecause of the direct dependence of the derivative on the linear structure ofC 0 [0, T ]. Considering the Gâteaux derivative, we have that for h ∈ C 0 [0, T ], theGateaux derivative of W t in direction h isW t (ω + εh) − W t (ω) ω t + εh t − ω tD h W t (ω) = lim= lim= h tε→0 εε→0 εThis means that this derivative is not amenable to direct generalization either, becausethe derivative is an element of the underlying space depending directly of its propertiesas a function space. Consider, however, the case where h t = ∫ tg(t) dt for some0g ∈ L 1 [0, T ]. Then,D h W t (ω) = h t =∫ T01 [0,t] (s)g(s) ds.In this case, then, the Gâteaux derivative for any ω ∈ C 0 [0, T ] is actually characterizedby the mapping 1 [0,t] from [0, T ] to R. We can therefore consider 1 [0,t] as a kind ofderivative of W t . Extending this concept to general mappings X : C 0 [0, T ] → R, wecan say that if there exists a mapping f : [0, T ] → R such that for h ∈ C 0 [0, T ] withh(t) = ∫ t0 g(s) ds, it holds that D hX(ω) = ∫ Tf(s)g(s) ds, then f is the derivative of0X, and we define D F X = f.To interpret this derivative, fix y ∈ [0, T ] and consider the case h ε (t) = ∫ t0 ψ ε(s−y) ds,where (ψ ε ) is a Dirac family, see Appendix A. This means that h ε is 0 almost all theway on [0, y] and 1 almost all the way on [y, T ]. Then,D hε X(ω) =∫ T0D F X(s)ψ ε (s − y) ds ≈ D F X(y).In other words, D F X(y) can be thought of as measuring the sensitivity of X to parallelshifts of the argument ω of the form 1 [y,T ] .

4.1 The spaces S and L p (Π) 129Led by the above ideas, consider now a general probability triple (Ω, F, P ) with aWiener process W on [0, T ]. By F T , we denote the usual augmentation of the σ-algebra generated by W . We want to define the Malliavin derivative of W t as themapping DW t = 1 [0,t] from [0, T ] to R. Since this derivative in the abstract setting isnot defined by a limiting procedure, but only by analogy to the case of C 0 [0, T ], weneed to impose requirements on the D to extend it meaningfully to other variablesthan W t . A suitable starting point is to require that a chain rule holds for D: Iff is continuously differentiable with sufficient growth conditions, namely polynomialgrowth of the mapping itself and its partial derivatives, we would like thatDf(W t1 , . . . , W tn ) =n∑k=1∂f∂x k(W t1 , . . . , W tn )DW tk ,where we put DW tk = 1 [0,tk ]. It turns out that to require this form of the derivative off(W t1 , . . . , W tn ) is sufficient to obtain a rich theory for the operator D. The derivativegiven above is a stochastic function from [0, T ] to R. With sufficient growth conditionson f, it is an element of L p ([0, T ] × Ω), p ≥ 1.Our immediate goal is to formalize the definition of the derivative given above andconsider its properties. To do so, we begin by accurately specifying the immediatedomain of the derivative operator and considering its properties. After this, we willrigorously define the derivative operator.4.1 The spaces S and L p (Π)We will now define the space S which will be the initial domain of the Malliavinderivative. We work in the context of a filtered probability space (Ω, F, P, F t ) with aone-dimensional Brownian motion W , where the filtration is the augmented filtrationinduced by the Brownian motion. A multi-index of order n is an element α ∈ N n 0 . Thedegree of the multi-index is |α| = ∑ nk=1 α k. A polynomial in n variables of degree kis a map p : R n → R, p(x) = ∑ |α|≤k a αx α . The sum in the above runs over all multiindiciesα with |α| ≤ k, with x α = ∏ ni=1 xα ii . The space of polynomials of degree k inany number of variables is denoted P k . Furthermore, we use the following notation.• C 1 (R n ) are the mappings f : R n → R which are continuously differentiable.• C ∞ (R n ) are the mappings f : R n → R which are differentiable infinitely often.

130 The Malliavin Calculus• C ∞ p (R n ) are the elements f ∈ C ∞ (R n ) such that f and its partial derivativesare dominated by polynomials.• C ∞ c(R n ) are the elements of C ∞ (R) with compact support.Clearly, then Cc∞ (R n ) ⊆ Cp ∞ (R n ) ⊆ C ∞ (R n ) ⊆ C 1 (R n ). For basic results on thesespaces, see appendix A. We are now ready to define the space S.Definition 4.1.1. If t ∈ [0, T ] n , we write W t = (W t1 , . . . , W tn ). By S, we then denotethe space of variables f(W t ), where f ∈ C ∞ p (R n ) and t ∈ [0, T ] n .Our aim in the following regarding S is to develop a canonical representation forelements of S, develop results to yield flexible representations of elements in S andto prove that S is an algebra which is dense in L p (F T ), p ≥ 1. These results aresomewhat tedious, but they are necessary for what will follow.Lemma 4.1.2. Let t ∈ [0, T ] n , s ∈ [0, T ] m and F ∈ S with F = f(W t ). Assume that{t 1 , . . . , t n } ⊆ {s 1 , . . . , s m }. There exists g ∈ C ∞ p (R m ) such that F = g(W s ).Comment 4.1.3 Note that this lemma not only allows us to extend the coordinateswhich F depends on, but also allows us to reorder the coordinates.◦Proof. Since {t 1 , . . . , t n } ⊆ {s 1 , . . . , s m }, we have n ≤ m and there exists a mappingσ : {1, . . . , n} → {1, . . . , m} such that t k = s σ(k) for k ≤ n. We then simply defineg(x 1 , . . . , x m ) = f(x σ1 , . . . , x σn ). Then g ∈ C ∞ p (R m ) andg(W s ) = g(W s1 , . . . , W sm )= f(W sσ1 , . . . , W sσ n )= f(W t1 , . . . , W tn )= f(W t )Lemma 4.1.4. Let F ∈ S with F = f(W t ) for f ∈ C ∞ p (R n ). If t 1 = 0, there existsg ∈ C ∞ p (R n−1 ) such that F = g(W t2 , . . . , W tn ).Proof. Define g(x 2 , . . . , x n ) = f(0, x 2 , . . . , x n ). Because we have W 0 = 0, it holds thatF = f(W t1 , . . . , W tn ) = g(W t2 , . . . , W tn ).

4.1 The spaces S and L p (Π) 131Lemma 4.1.5. Let F ∈ S with F = f(W t ) for f ∈ C ∞ p (R n ). If t i = t j for some i ≠ j,there is g ∈ C ∞ p (R n−1 ) such that F = g(W u ), where u = (t 1 , . . . , t j−1 , t j+1 . . . , t n ).Proof. Define A : R n−1 → R n by Ax = (x 1 , . . . , x j−1 , x i , x j+1 , . . . , x n ). Then A islinear. Putting g(x) = f(Ax), then g ∈ C ∞ p (R n ). Clearly, AW u = W t and thereforeF = f(W t ) = f(AW u ) = g(W u ), as desired.Corollary 4.1.6. Let F ∈ S. There exists t ∈ [0, T ] n such that 0 < t 1 < · · · < t n andf ∈ C ∞ p (R n ) such that F = f(W t ).Proof. This follows by combining Lemma 4.1.2, Lemma 4.1.4 and Lemma 4.1.5.Corollary 4.1.6 states that any element of S has a representation where we can assumethat all the coordinates of the Wiener process in the element are different, positiveand ordered. This observation will make our lives a good deal easier in the following.Our next result shows that any element of S can be written as a transformation of ann-dimensional standard normal variable.Lemma 4.1.7. Let F ∈ S with F = f(W t ) for f ∈ Cp ∞ (R n ), where 0 < t 1 < · · · < t n .Define A : R n → R n by Ax = ( √ x1t1, √ x2−x1t2−t 1, . . . , √ xn−xn−1tn −t n−1). Then A is invertible,f ◦ A −1 ∈ Cp ∞ (R n ) and F = (f ◦ A −1 )(AW t ), where AW t has the standard normaldistribution in n dimensions.Proof. Put t 0 = 0. A is invertible with inverse given by (A −1 x) k = ∑ ki=0√tk − t k−1 x k .Since A −1 is linear, f ◦ A −1 ∈ C ∞ p (R n ). It is clear that AW t is standard normal.We are now done with our results on representations of elements of S. Before proceeding,we prove that S is an algebra which is dense in L p (F T ).Lemma 4.1.8. S is an algebra, and S ⊆ L p (Ω) for all p ≥ 1.Proof. S is an algebra. S is a space of real random variables. Since S is obviouslynonempty, to prove the first claim, we need to show that S is stable under linearcombinations and multiplication. Let F, G ∈ S with F = f(W t ) and G = g(W s ) wheref ∈ C ∞ p (R n ) and g ∈ C ∞ p (R m ). Let λ, µ ∈ R. Then,λF + µG = (λf ⊕ µg)(W t1 , . . . , W tn , W s1 , . . . , W sm ),

132 The Malliavin Calculuswhere λf ⊕ µg ∈ C ∞ p (R n+m ). In the same manner,F G = (f ⊗ g)(W t1 , . . . , W tn , W s1 , . . . , W sm ),where, f ⊗ g ∈ C ∞ p (R n+m ). This shows that S is an algebra.S is a subset of L p (Ω). Next, we will show that S ⊆ L p (Ω). Let F ∈ S be suchthat F = f(W t ). By Corollary 4.1.6, we can assume 0 < t 1 < · · · < t n . By Lemma4.1.7, there then exists g ∈ C ∞ p (R n ) such that F = g(X) where X is n-dimensionallystandard normally distributed. Since g ∈ C ∞ p (R n ), |g| p ∈ C ∞ p (R n ). Therefore, thereis a multinomial q in n variables with |g| p ≤ q, say q(x) = ∑ |α|≤m a αx α . We then findE|F | p =≤≤∫∫∫= ∑|g(x)| p φ n (x) dxq(x)φ n (x) dx∑n∏|α|≤m k=1|α|≤m k=1n∏∫a αa α x α kkφ n(x) dxx α kk φ(x k) dx k ,which is finite, since the normal distribution has moments of all orders.Lemma 4.1.9. The variables of the form f(W t ) for f ∈ C ∞ c (R n ) are dense in L p (F T )for p ≥ 1.Proof. As in the proof of Lemma 3.7.5, we find that with G T the σ-algebra generated byW , F T ⊆ σ(G T , N ), so it will suffice to show that the variables given in the statementof the lemma is dense in L 2 (G T ).We first prove that any X ∈ L p (G T ) can be approximated by variables dependentonly on finitely many coordinates of the Wiener process. Let X ∈ L p (G T ) be givenand let {t k } k≥1 be dense in [0, T ] and let H n = σ(W t1 , . . . , W tn ). Then H n is an increasingfamily of σ-algebras, and G T = σ(H n ) n≥1 . Therefore, E(X|H n ) is a discretetimemartingale with limit X, convergent in L p . By the Doob-Dynkin lemma B.6.7,E(X|H n ) = g(W t1 , . . . , W tn ) for some measurable g : R n → R.Now consider X ∈ L p (G T ) with X = g(W t ) for some measurable g : R n → R andt ∈ [0, T ] n . Let µ be the distribution of W t . Since µ is bounded, µ is a Radon measure

4.1 The spaces S and L p (Π) 133on R n , and g ∈ L p (µ) by assumption. By Theorem A.4.5, g can be approximated inL p (µ) by mappings (g n ) ⊆ Cc∞ (R n ). We then obtainshowing the claim of the lemma.lim ‖g(W t ) − g n (W t )‖ p = lim ‖g − g n ‖ L p (µ) = 0,Corollary 4.1.10. The space S is dense in L p (F T ) for any p ≥ 1.Proof. This follows immediately from Lemma 4.1.9, since C ∞ c (R n ) ⊆ C ∞ p (R n ).This concludes our basic results on S, which we will use in the following sections whendeveloping the Malliavin derivative. The important points of our work is this:• By Corollary 4.1.6, we can choose a particularly nice form of any F ∈ S, namelythat F = f(W t ) where 0 < t 1 < · · · < t n . Also, by Lemma 4.1.2 we can extendto more coordinates when we wish. When working with more than one elementof S, this will allow us to assume that they depend on the same underlyingcoordinates of the Wiener process.• By combining Corollary 4.1.6 and Lemma 4.1.7, we can always write any F ∈ Sas a Cp∞ transformation of a standard normal variable.• By Lemma 4.1.8 and Corollary 4.1.10, S is an algebra, dense in L p (F T ) for anyp ≥ 1.Before we proceed to the next section, we show some basic results on the spaceL p ([0, T ] × Ω, B[0, T ] ⊗ F T , λ ⊗ P ). We will use the shorthands Π = B[0, T ] ⊗ F Tand η = λ ⊗ P and write L p (Π) for L p ([0, T ] × Ω, B[0, T ] ⊗ F T , λ ⊗ P ). When p = 2,we denote the inner product on L 2 (Π) by 〈·, ·〉 Π .Lemma 4.1.11. The variables of the form f(W t ) for f ∈ C ∞ c (R n ) generate F T .Proof. This follows immediately from Lemma 4.1.9, since this lemma yields that forany A ∈ F T , 1 A can be approximated pointwise almost surely with elements of theform f(W t ), where f ∈ C ∞ c (R n ).Lemma 4.1.12. The functions 1 [s,t] ⊗ f(W u ), where 0 ≤ s ≤ t ≤ T , u ∈ [0, T ] n andf ∈ C ∞ c (R n ) is stable under multiplication and generates Π.

134 The Malliavin CalculusProof. Clearly, the class is stable under multiplication. We show the statement aboutits generated σ-algebra. We will apply Lemma B.6.4. Let E be the class of mappingsf(W u ) where f ∈ Cc∞ (R n ) and u ∈ [0, T ] n . Let K be the class of functions 1 [s,t] for0 ≤ s ≤ t ≤ T .By Lemma 4.1.11, σ(E) = F T . Since f(W 0 ) is constant for any f ∈ Cc∞ (R), E containsthe constant mappings. Also, clearly σ(K) = B[0, T ], and clearly the mapping whichis constant 1 on [0, T ] is in K. The hypotheses of Lemma B.6.4 are then fulfilled, andthe conclusion follows.4.2 The Malliavin Derivative on S and D 1,pWith the results of Section 4.1, we are ready to define and develop the Malliavinderivative. Letting F ∈ S with F = f(W t ), where f ∈ C ∞ p (R n ), we want to putDF =n∑k=1∂f∂x k(W t )1 [0,tk ].We begin by showing that this is well-defined, in the sense that if we have differentrepresentations f(W t ) = g(W s ), the derivative yields the same result.Lemma 4.2.1. Let F ∈ S, and assume that we have two representations,F = f 1 (W t )F = f 2 (W s ),where f 1 ∈ C ∞ p (R n ) and f 2 ∈ C ∞ p (R m ). It then holds thatn∑k=1∂f 1∂x k(W t )1 [0,tk ] =m∑k=1∂f 2∂x k(W s )1 [0,sk ].Proof. We proceed in two steps, first showing the equality for particularly simplerepresentations of F and then proceeding to the general case.Step 1. First assume that F = f 1 (W t ) = f 2 (W t ) for some f 1 , f 2 ∈ Cp ∞ (R n ), where0 < t 1 < · · · < t n . Define A : R n → R n by Ax = ( √ x 1t1, √t2−t x 2−x 11, . . . , x n−x n−1√tn−t n−1).By Lemma 4.1.7, A is invertible, and AW t has the standard normal distribution inn dimensions. Put g 1 = f 1 ◦ A −1 and g 2 = f 2 ◦ A −1 . Then g 1 (AW t ) = g 2 (AW t ).

4.2 The Malliavin Derivative on S and D 1,p 135This implies that g 1 and g 2 are equal almost surely with respect to the n-dimensionalstandard normal measure. Since this measure has a positive density with respect tothe Lebesgue measure, this means that g 1 and g 2 are equal Lebesgue almost surely.Since both mappings are continuous, we conclude g 1 = g 2 . Because f 1 = g 1 ◦ A andf 2 = g 2 ◦ A, this implies f 1 = f 2 and thereforeas desired.n∑k=1∂f 1∂x k(W t )1 [0,tk ] =n∑k=1∂f 2∂x k(W t )1 [0,tk ],Step 2. Now assume F = f 1 (W t ) and F = f 2 (W s ). Defining u = (t 1 , . . . , t n , s 1 , . . . , s m )and puttingg 1 (x 1 , . . . , x n+m ) = f 1 (x 1 , . . . , x n )g 2 (x 1 , . . . , x n+m ) = f 2 (x n+1 , . . . , x m ),we obtain f 1 (W t ) = g 1 (W u ) and f 2 (W s ) = g 2 (W u ). By what we already have shown,Therefore, we concluden+m∑k=1n+m∂g 1∑ ∂g 2(W u )1 [0,tk ] = (W u )1 [0,tk ].∂x k ∂x kk=1n∑k=1∂f 1∂x k(W t )1 [0,tk ] ==n∑k=1n+m∑k=1n+m∂g 1∑ ∂g 1(W u )1 [0,uk ] = (W u )1 [0,uk ]∂x k ∂x kk=1m∑∂g 2∂x k(W u )1 [0,uk ] =k=1∂f 2∂x k(W s )1 [0,sk ],showing the desired result.Definition 4.2.2. Let F ∈ S, with F = f(W t1 , . . . , W tn ). We define the Malliavinderivative of F as the stochastic process on [0, T ] given byn∑k=1∂f∂x k(W t1 , . . . , W tn )1 [0,tk ].Comment 4.2.3 The mapping defined above is well-defined by Lemma 4.2.1.◦We have now defined the Malliavin derivative D on S. Next, we will investigate itsbasic properties and extend it to a larger space.

136 The Malliavin CalculusLemma 4.2.4. The mapping D is linear and maps into L p (Π) for all p ≥ 1.Proof. Let F, G ∈ S with F = f(W t ) and G = g(W s ). Also, let λ, µ ∈ R. FromLemma 4.1.8 and its proof, we haveD(λF + µG) = D(λf ⊕ µg)(W t1 , . . . , W tn , W s1 , . . . , W sm )=n∑λ ∂fm∑(W t )1 [0,tk ] + µ ∂g (W s )1 [0,sk ]∂x k ∂x kk=1= λDF + µDG.This shows linearity. To show the second statement, assume F = f(W t ). Then,n∑ ∂fn∑‖DF ‖ p =(W∥t )1 [0,tk ]≤ T 1 p∂f∂x k ∥ ∥ (W t )∂x k∥ .pSince f ∈ Cp ∞ (R n ),lemma.∂f∂x kk=1pk=1k=1has polynomial growth, so ∂f∂x k(W t ) ∈ L p (F T ), proving theFrom Corollary 4.1.10, we see that we have defined the derivative on a dense subspaceof L p (F T ). Nonetheless, even dense sets can be quite slim, and we need to extend theoperator D to a larger space before it can be of any actual use. To do so, we showthat D is closable. We can then extend it by taking the closure. We first need a fewlemmas.Lemma 4.2.5. Let F ∈ S with F = f(W t ) for f ∈ C ∞ p (R n ), where t = (t 1 , . . . , t n )and 0 = t 0 < t 1 < · · · < t n . Let A : R n → R n be linear and invertible with matrix A ′ .Thenn∑k=1∂∂x kf(W t )1 [0,tk ] =n∑n∑k=1 i=1∂∂x i(f ◦ A −1 )(AW t )A ′ ik1 [0,tk ].Proof. This follows fromn∑ ∂f(W t )1 [0,tk ] =∂x kk=1==n∑k=1n∑k=1 i=1n∑∂∂x k(f ◦ A −1 ◦ A)(W t )1 [0,tk ]n∑n∑k=1 i=1where we have applied the chain rule.∂(f ◦ A −1 )(AW t ) ∂ A i (W t )1 [0,tk ]∂x i ∂x k∂∂x i(f ◦ A −1 )(AW t )A ′ ik1 [0,tk ],

4.2 The Malliavin Derivative on S and D 1,p 137Lemma 4.2.6. Let f ∈ Cp ∞ (R n ). Let φ n be the n-dimensional standard normaldensity. Then,∫∫∂f(x)φ n (x) dx =∂x kf(x)x k φ n (x) dx.Proof. First note that the integrals are well-defined, since φ n decays faster than anypolynomial. We begin by showing that it will suffice to prove the result in the casewhere k = 1, this will make the proof notationally easier. To this end, note that theLebesgue measure is invariant under orthonormal transformations. Define the mappingU : R n → R n as the linear transformation exchanging the first and k’th coordinates,leaving all other coordinates the same. U is then orthonormal, and∫∂f∂x k(x)φ n (x) dx ===∫ ∂f(Ux)φ n (Ux) dx∂x k∫∑ n∂f(Ux) ∂U i(x)φ n (Ux) dx∂xi=1 i ∂x 1∫ ∂(f ◦ U)(x)φ n (x) dx.∂x 1Likewise,∫∫f(x)x k φ n (x) dx =∫f(Ux)(Ux) k φ n (Ux) dx =(f ◦ U)(x)x 1 φ n (x) dx.Therefore, it will suffice to show the result for k = 1 for mappings of the form f ◦ Uwhere f ∈ C ∞ p (R n ). But such mappings are in C ∞ p (R n ) as well, and we conclude thatit will suffice to show the result in the case where k = 1 for some f ∈ C ∞ p (R n ). Inthat case, we obtain∫= limM→∞= limM→∞∫= limM→∞∂f(x)φ n (x) dx∂x 1∫[−M,M] n∫ M∫[−M,M] n−1∂f∂x 1(x)φ n (x) dx−M∂f∂x 1(x)φ n (x) dx 1 · · · dx n∫ M[f(x)φ n (x)] x1=Mx 1 =−M −[−M,M] n−1 −Mf(x) ∂φ n∂x 1(x) dx 1 dx 2 · · · dx n .We wish to use dominated convergence to get rid of the first term. Defining x ′ by

138 The Malliavin Calculusx ′ = (x 2 , . . . , x n ), we obtain∫ ∣ ∣∣[f(x)φn(x)] x 1=Mx 1=−M ∣ dx 2 · · · dx n[−M,M]∫ ∣ n−1 ∣∣[f(x)φn≤(x)] x1=Mx 1 =−M ∣ dx ′∫≤ |f(M, x ′ )|φ n (M, x ′ ) + |f(−M, x ′ )|φ n (−M, x ′ ) dx ′∫∫= |f(M, x ′ )|φ(M)φ n−1 (x ′ ) dx ′ + |f(−M, x ′ )|φ(−M)φ n−1 (x ′ ) dx ′Let p with p(x) = ∑ mk=0 a k∏ ni=1 xαk i , be a polynomial such that |f| ≤ p. Then,sup |f(M, x ′ )|φ(M)M∈R≤=m∑n∏|a k | sup φ(M)M αk 1|x| αk iM∈Ri=2m∑()∏ n|a k | sup |M| αk 1 φ(M) |x| αk i ,M∈Rk=0k=0i=2where the suprema are finite since φ decays faster than any exponential. Therefore,sup M∈R |f(M, x ′ )|φ 1 (M) has polynomial growth. From this we conclude thatsup M∈R |f(M, x ′ )|φ 1 (M)φ n−1 (x ′ ) is integrable, and by dominated convergence, wetherefore get==limM→∞∫∫= 0.∫[−M,M] n−1 [f(x)φ n (x)] x 1=Mx 1=−M dx′lim 1 [−M,M]M→∞ n−1(x′ )(fφ n )(M, x ′ ) − 1 [−M,M] n−1(x ′ )(fφ n )(−M, x ′ ) dx ′lim (fφ n)(M, x ′ ) − (fφ n )(−M, x ′ ) dx ′M→∞We may therefore conclude∫∂f(x)φ n (x) dx = − lim∂x 1∫= lim=M→∞M→∞∫∫[−M,M] n−1 ∫ M−M[−M,M] n f(x)x 1 φ n (x) dxf(x)x 1 φ n (x) dx,f(x) ∂φ n∂x 1(x) dx 1 dx 2 · · · dx nas desired.

4.2 The Malliavin Derivative on S and D 1,p 139Comment 4.2.7 The result of Lemma 4.2.6 is a version of a result sometimes knownas Stein’s lemma in the literature, see for example Zou et al. (2007). See Stein (1981),Lemma 1 and Lemma 2 for proofs of Stein’s Lemma.◦Lemma 4.2.8. Consider G = g(W u ) ⊗ 1 [s,t] with g ∈ Cc∞ (R n ) and u ∈ [0, T ] n . Thereis some constant C such that for any F ∈ S, | ∫ G(DF ) dη| ≤ C‖F ‖ p .Proof. By Lemma 4.1.2, we can assume that F = f(W u ) for some f ∈ Cp ∞ (R n ), andwe can assume that 0 < u 1 < · · · < u n . Define Ax = ( √ x1 xu1, √u2 2−x 1 x−u 1, . . . , √un−u n−x n−1n−1),by Lemma 4.2.5 we then haven∑k=1∂∂x kf(W u )1 [0,uk ] =n∑n∑k=1 i=1∂∂x i(f ◦ A −1 )(AW u )A ′ ik1 [0,uk ],where A ′ is the matrix for the linear mapping A. Defining f ′ = f ◦A −1 and g ′ = g◦A −1 ,we then obtain∫GDF dη ===∫ n∑ n∑ ∂f ′G(AW u )A ′∂xik1 [0,uk ] dηk=1 i=1 in∑ n∑∫A ′ ik g(W u ) ∂f ′(AW u )1 [s,t] 1 [0,uk ] dη∂xk=1 i=1in∑ n∑(∫)A ′ ik 1 [s,t] 1 [0,uk ] dλ E(g ′ (AW u ) ∂f ′ )(AW u ) .∂x ik=1 i=1In particular, then,∫∣GDF dη∣ ≤ nT ‖A′ ‖ ∞n∑i=1(∣ E g ′ (AW u ) ∂f )∣ ′ ∣∣∣(AW u ) .∂x iWe consider the mean values in the sums. Using Lemma 4.2.6, we obtainE(g ′ (AW u ) ∂f ′ )(AW u )∂x i===∫g ′ (x) ∂f ′(x)φ n (x) dx∂x i∫ ∂g ′ f ′∫ ∂g′(x)φ n (x) dx − (x)f ′ (x)φ n (x) dx∂x i∂x∫∫ i∂gg ′ (x)f ′ ′(x)x i φ n (x) dx − (x)f ′ (x)φ n (x) dx.∂x i

140 The Malliavin CalculusRecalling that F = f ′ (AW u ), we can use Hölder’s inequality to obtain(∣ E g ′ (AW u ) ∂f )∣ ′ ∣∣∣(AW u ) =∂x i=≤∫∫ ∂g∣ g ′ (x)f ′ ′(x)x i φ n (x) dx − (x)f ′ (x)φ n (x) dx∂x i∣(∣ EF g ′ (AW u )(AW u ) i − ∂g′ ∣∣∣(AW u ))∣∂x i∥ ∥∥∥‖F ‖ p g ′ (AW u )(AW u ) i − ∂g′(AW u )∂x i∥ .qSince g has compact support, so does g ′ and ∂g′∂x i. Therefore, the latter factor in theabove is finite. PuttingC = nT ‖A ′ ‖ ∞n∑i=1we have proved | ∫ GDF dη| ≤ C‖F ‖ p , as desired.∥ g′ (AW u )(AW u ) i − ∂g′(AW u )∂x i∥ ,qTheorem 4.2.9. The operator D is closable from S to L p (Π).Proof. Let a sequence (F n ) ⊆ S be given such that F n converges to zero in L p (F T )and such that DF n converges in L p (Π). We need to prove that the limit of DF n iszero. Let ξ be the limit, By Theorem A.2.5 it will suffice to prove that that for anyG ∈ L q (Π), ∫ ξG dη = 0.Let H be the class of elements G ∈ L q (Π) such that ∫ ξG dη = 0. We will provethat H = L q (Π) first by proving that bH contains all bounded mappings in L q (Π).Obviously, then, H also contains all bounded mappings in L q (Π) and we can finish theproof by a closedness argument.Step 1: bH contains all bounded mappings. We will use the monotone classtheorem to prove that bH contains all bounded mappings. To this end, note thatbH by dominated convergence is a monotone vector space. Defining K as the classof mappings g(W t ) ⊗ 1 [s,t] where g ∈ Cc∞ (R n ) and t ∈ [0, T ] n , we know from Lemma4.1.12 that K is a multiplicative class generating Π. If we can show that K ⊆ bH,it will follow from the monotone class theorem B.1.2 that bH contains all boundedmappings in L q (Π).Let G ∈ K be given with G = g(W u ) ⊗ 1 [s,t] . Our ambition is to show G ∈ H.By Lemma 4.2.8, there exists a constant C, such that for any F ∈ S, it holds that

4.2 The Malliavin Derivative on S and D 1,p 141| ∫ G(DF ) dη| ≤ C‖F ‖ p . By the continuity of F ↦→ ∫ F G dη, we then obtain∫∣∫∫∣∣∣ ∣ ξG dη∣ = G lim DF n dηn ∣ = lim n ∣ G(DF n ) dη∣ ≤ lim C‖F n‖ p = 0.nThus, ∫ Gξ dη = 0. We conclude G ∈ bH and therefore K ⊆ H.Step 2: H is closed. Since the bounded elements of L q (Π) are dense in L q (Π), itwill follow that H = L q (Π) if we can prove that H is closed. To do so, assume that(G n ) ⊆ H, converging to G. Then lim ‖ξG n − ξG‖ 1 ≤ lim ‖ξ‖ p ‖G n − G‖ q = 0 byHölder’s inequality, so ∫ ξG dη = lim ∫ ξG n dη = 0, and H is closed. It follows thatH = L q (Π) and we may finally conclude that ξ = 0 and therefore D is closable.We may now conclude from Theorem A.7.3 that D has a unique closed extension fromS to the subspace D 1,p of L p (F T ) defined as the F ∈ L p (F T ) such that there existsa sequence (F n ) in S converging to F such that DF n converges as well, and in thiscase the value of DF is the limit of DF n . D maps from D 1,p into L p (Π). Note thatendowing the space D 1,p with the norm given by ‖F ‖ 1,p = ‖F ‖ p +‖DF ‖ p , S is a densesubspace of D 1,p . This is another way to think of the extension of D from S to D 1,p .We will mainly be interested in the properties of the operator from D 1,2 → L 2 (Π). Wewill consider this case in detail in the next section. First, we will spend a momentreviewing our progress. We have succeeded in defining, for any p ≥ 1, a linear operatorD : D 1,p → L p (Π) with the properties that• For F ∈ S with F = f(W t ), DF = ∑ n ∂fk=1 ∂x k(W t )1 [0,tk ].• When considering D 1,p under the norm ‖ · ‖ p , D is closed.These two properties are more or less all there is to the definition. The first propertyshows how to calculate D on S, and the second property shows how to infer valuesof D outside of S from the values on S. Note that our definition of D 1,p is differentfrom the one seen in Nualart (2006). Nualart (2006) consideres the construction ofthe Malliavin derivative based on isonormal gaussian processes, which can be seen asa generalization of Brownian motion. This allows certain proofs the clarity of higherabstraction, but the downside is that the theory becomes harder to relate to realproblems and furthermore that the development of the derivative operator is tied upto the L 2 -structure of the concept of isonormal processes. For a brownian motion, the

142 The Malliavin Calculusspace of smooth functionals considered in Nualart (2006) would be{f( ∫ T0∫ }T∣∣∣∣h 1 (t) dW (t), . . . , h n (t) dW (t))∣h 1 , . . . , h n ∈ L 2 [0, T ], f ∈ Cp ∞ (R n ) ,0and the space D 1,p would be the completion of this space with respect to the norm‖F ‖ 1,p =()E|F | p + E‖DF ‖ p 1pL 2 [0,T ],where DF is considered as a stochastic variable with values in L 2 [0, T ].4.3 The Malliavin Derivative on D 1,2We will now consider specifically the properties of the Malliavin derivative as operatorD : D 1,2 → L 2 (Π). We will prove two results allowing us easy manipulation of theMalliavin derivative: the chain rule and the integration-by-parts formula. Furthermore,we will prove that D 1,2 has some degree of richness by showing that it containsall stochastic integrals of deterministic functions. We will also show how the derivativeof such integrals are given.We begin by proving the chain rule. The proof of the general case is rather long, so wesplit it in two. We begin by proving in Theorem 4.3.1 the chain rule as it is formulatedin Nualart (2006) with transformations having bounded partial derivatives. Afterthis, we prove in Theorem 4.3.5 a generalized chain rule.Theorem 4.3.1. Let F = (F 1 , . . . , F n ), where F 1 , . . . , F n ∈ D 1,2 , and let ϕ ∈ C 1 (R n )with bounded partial derivatives. Then ϕ(F ) ∈ D 1,2 andDϕ(F ) =n∑k=1∂ϕ∂x k(F )DF k .∂ϕComment 4.3.2 In the above,∂x k(F ) is a mapping on Ω and DF k is a mapping on[0, T ] × Ω. The multiplication is understood to be pointwise in the sense( ∂ϕ∂x k(F )DF k)(t, ω) = ∂ϕ∂x k(F )(ω)DF k (t, ω).◦

4.3 The Malliavin Derivative on D 1,2 143Proof. The conclusion is well-defined since ∂ϕ∂x kis bounded, so ϕ(F ) is an element ofL 2 (F T ) and the right-hand side is an element of L 2 (Π). We prove the theorem in threesteps:1. The case where ϕ ∈ C ∞ p (R n ) and F ∈ S n .2. The case where ϕ ∈ C ∞ p (R n ) with bounded partial derivatives and F ∈ D n 1,2.3. The case where ϕ ∈ C 1 (R n ) with bounded partial derivatives and F ∈ D n 1,2.Step 1: Cp∞ transformations of S. Consider the case where ϕ ∈ Cp ∞ (R n ) andF ∈ S n . By Lemma 4.1.2, we can assume that the F k are transformations of thesame coordinates of the wiener process, F k = f k (W t ) where t ∈ [0, T ] m . Puttingf = (f 1 , . . . , f n ), ϕ ◦ f ∈ Cp ∞ (R m ). We have ϕ(F ) = (ϕ ◦ f)(W t ), so ϕ(F ) ∈ S andthereforem∑ ∂(ϕ ◦ f)Dϕ(F ) =(W t )1 [0,ti ]∂x i===i=1m∑n∑i=1 k=1n∑k=1n∑k=1∂ϕ∂x k(f(W t )) ∂f k∂x i(W t )1 [0,ti]∂ϕ∂x k(f(W t ))∂ϕ∂x k(F )DF k .m∑i=1∂f k∂x i(W t )1 [0,ti]Step 2: Special Cp∞ transformations of D 1,2 . We next consider the case whereF ∈ D n 1,2 and ϕ ∈ Cp ∞ (R n ) with bounded partial derivatives. Since D is closed, weknow that there exists F j k ⊆ S such that F j k converges to F k in L 2 (F T ) and DF j kconverges to DF k in L 2 (Π). Then, by what we have already shown, ϕ(F j ) ∈ S andDϕ(F j ) = ∑ n ∂ϕk=1 ∂x k(F j )DF j k . Our plan is to show that ϕ(F j ) converges to ϕ(F ) inL 2 (F T ) and that Dϕ(F j ) converges to ∑ n ∂ϕi=1 ∂x i(F i )DF i in L p (Π). By the closednessof D, this will imply that ϕ(F ) ∈ D 1,2 and Dϕ(F ) = ∑ n ∂ϕk=1 ∂x k(F )DF k .By assumption, F j k converges to F k in L 2 (F T ) and DF j k converges in L2 (Π) to DF k .By picking a subsequence we may assume that we also have almost sure convergence.By the mean value theorem, Lemma A.2.2,∥∥ϕ(F j ) − ϕ(F ) ∥ ∥2=n∑ ∂ϕ(ξ)(F j∥k∂x − F k)k ∥k=12≤ maxk≤n∥ ∥ ∂ϕ ∥∥∥∞ ∥∥∥∥ ∑n ∥(F j k∂x − F k) ∥ ,kk=1∥2

144 The Malliavin Calculusso ϕ(F j ) converges to ϕ(F ) in L 2 (F T ). To show the other convergence, note that∥ Dϕ(F j ) −n∑i=1∂ϕ∂x i(F )DF i∥ ∥∥∥∥2=≤∥ n∑ ∂ϕ(F j )DF j i∥ ∂x − ∑n ∂ϕ ∥∥∥∥2(F )DF ii=1 i ∂xi=1 in∑∂ϕ∥ (F j )DF j i∂x − ∂ϕ ∥ ∥∥∥2(F )DF i ,i ∂x iwhere, for each i ≤ n,∂ϕ∥ (F j )DF j i∂x − ∂ϕ ∥ ∥∥∥2(F )DF ii ∂x i ( ≤∂ϕ∥ (F j )(DF j i∂x − DF i)i∥ +∂ϕ∥ (F j ) − ∂ϕ ) ∥ ∥∥∥2(F ) DF i2∂x i ∂x i ∥ ≤∂ϕ ∥∥∥∞ ∥ (∥∥DF j∥i∂x − DF i∥ +∂ϕi 2∥ (F j ) − ∂ϕ ) ∥ ∥∥∥2(F ) DF i .∂x i ∂x ii=1Here, the first term converges to zero by assumption. Regarding the second term, weknow that F j −→ a.s.∂ϕF , and by continuity,∂x i(F j ) −→ a.s. ∂ϕ∂x i(F ), bounded by a constantsince the partial derivatives are bounded. Dominated convergence therefore yields thatthe second term in the above converges to zero as well. We conclude that ϕ(F ) ∈ D 1,2and Dϕ(F ) = ∑ n ∂ϕk=1 ∂x k(F )DF k .Step 3: C 1 transformations of D 1,2 . Now assume ϕ ∈ C 1 with bounded partialderivatives, and let F ∈ D 1,2 . From Lemma A.4.3, we know that there exists functionsϕ n ∈ Cp ∞ (R n ) such that ϕ n converges uniformly to ϕ, the partial derivatives of ϕ nconverges pointwise to the partial derivatives of ϕ and ‖ ∂ϕ n∂x k‖ ∞ ≤ ‖ ∂ϕ∂x k‖ ∞ . By whatwe already have shown, ϕ n (F ) ∈ D 1,2 and Dϕ n (F ) = ∑ n ∂ϕ nk=1 ∂x k(F )DF k . We wantto argue that ϕ n (F ) converges to ϕ(F ) in L 2 (F T ) and that Dϕ n (F ) converges to∑ n ∂ϕk=1 ∂x k(F )DF k in L 2 (Π). As in the previous step, by closedness of D, this will yieldthat ϕ(F ) ∈ D 1,2 and that the derivative is given by Dϕ(F ) = ∑ n ∂ϕk=1 ∂x k(F )DF k .Clearly, since ‖ϕ(F ) − ϕ n (F )‖ 2 ≤ ‖ϕ − ϕ n ‖ ∞ , ϕ n (F ) converges to ϕ(F ). To show thesecond limit statement, we noteSince ∂ϕ n∂x k∥ Dϕ n(F ) −n∑k=1∥∂ϕ ∥∥∥∥(F )DF k ≤∂x k2n∑( ∂ϕn∥ (F ) − ∂ϕ ) ∥ ∥∥∥2(F ) DF k .∂x k ∂x kk=1converges pointwise to ∂ϕ∂x k, clearly ∂ϕ n∂x k(F ) a.s.−→ ∂ϕ∂x k(F ), and this convergenceis bounded. The dominated convergence theorem therefore yields that the abovesum converges to zero.

4.3 The Malliavin Derivative on D 1,2 145The chain rule of Theorem 4.3.1 is sufficient in many cases, but it will be convenientin the following to have an extension which does not require the boundedness of thepartial derivatives of the transformation, but only requires enough integrability toensure that the statement of the chain rule is well-defined. This is what we now setout to prove.Lemma 4.3.3. Let M > 0 and ε > 0. There exists a mapping f ∈ Cc∞ (R n ) withpartial derivatives bounded by 1 + ε such that [−M, M] n ≺ f ≺ (−(M + 1), M + 1) n .Proof. Let ε > 0 be given, put M ε− = M − ε and M ε+ = M + ε. Define the setsK ε = [−M ε+ , M ε+ ] n and V ε = (−(M ε− + 1), M ε− + 1) n . We will begin by identifyinga d ∞ -Lipschitz mapping g ∈ C c (R n ) with K ε ≺ g ≺ V ε .To this end, we put g(x) = min{ 11−2ε d ∞(x, Vε c ), 1}. Clearly, then g is continuous andmaps into [0, 1]. Since g is zero on Vεc and V ε has compact closure, g has compactsupport. And if x ∈ K ε , we obtain{ }1g(x) = min1 − 2ε d ∞(x, Vε c ), 1{ }1≥ min1 − 2ε d ∞(K ε , Vε c ), 1 = 1,We conclude K ε ≺ g ≺ V ε . We know that |d ∞ (x, V c ) − d ∞ (y, V c )| ≤ d ∞ (x, y) by theinverse triangle inequality, so x ↦→ d ∞ (x, V c ) is d ∞ -Lipschitz continuous with Lipschitzconstant 1. Therefore, the mapping x ↦→ 11−2ε d ∞(x, V c ) is d ∞ -Lipschitz continuouswith Lipschitz constant11−2ε . Since x ↦→ min{x, 1} is a contraction, g is d ∞-Lipschitzcontinuous with Lipschitz constant11−2ε .Now let (ψ ε ) be a Dirac family with respect to ‖·‖ ∞ and let f = g∗ψ ε . We claim that fsatisfies the properties in the lemma. Let K = [−M, M] n and V = (−(M +1), M +1) n ,we want to show that f ∈ Cc∞ (R n ), that K ≺ f ≺ V and that f has partial derivatives1bounded by1−2ε. By Lemma A.3.2, f is infinitely differentiable, and it is clear thatf takes values in [0, 1]. If x ∈ V c , we have x − y ∈ Vε c for any y with ‖y‖ ∞ ≤ ε.Therefore,∫∫f(x) = g(x − y)ψ ε (y) dy = 1 (‖y‖∞ ≤ε)g(x − y)ψ ε (y) dy = 0.Likewise, if x ∈ K, then x − y ∈ K ε for any y with ‖y‖ ∞ ≤ ε and thus∫∫f(x) = g(x − y)ψ ε (y) dy = 1 (‖y‖∞≤ε)g(x − y)ψ ε (y) dy = 1.We have now shown K ≺ f ≺ V . Since V is compact closure, it follows in particularthat f has compact support. It remains to prove the bound on the partial derivatives.

146 The Malliavin CalculusTo this end, we note|f(x) − f(y)| =≤∫∫∣ g(x − z)ψ ε (z) dz − g(y − z)ψ ε (z) dz∣∫|g(x − z) − g(y − z)|ψ ε (z) dz≤ d ∞(x, y)1 − 2ε .We therefore obtain, with e k denoting the unit vector in the k’th direction,∂f∣ (x)∂x k∣ = lim |f(x + he k ) − f(x)|1 ‖he k ‖ ∞≤ lim= 1h→0 + hh→0 + 1 − 2ε h 1 − 2ε .We have now identified a mapping f ∈ Cc∞ (R n ) with K ≺ f ≺ V and partial derivativesbounded by11−2ε . Since ε > 0 was arbitrary, we conclude that there existsf ∈ Cc ∞ (R n ) with K ≺ f ≺ V and partial derivatives bounded by 1 + ε for any ε > 0.This shows the lemma.Comment 4.3.4 Instead of explicitly constructing the Lipschitz mapping, we couldalso have applied the extended Urysohn’s Lemma of Theorem C.2.2. This would haveyielded a weaker bound for the partial derivatives, but the nature of the argumentwould be more general.◦Theorem 4.3.5 (Chain rule). Let F = (F 1 , . . . , F n ), where F 1 , . . . , F n ∈ D 1,2 , andlet ϕ ∈ C 1 (R n ). Assume ϕ(F ) ∈ L 2 (F T ) and ∑ n ∂ϕk=1 ∂x k(F )DF k ∈ L 2 (Π). Thenϕ(F ) ∈ D 1,2 and Dϕ(F ) = ∑ n ∂ϕk=1 ∂x k(F )DF k .Proof. We first prove the corollary in the case where ϕ is bounded, and then extendto general ϕ.Step 1: The bounded case. Assume that ϕ ∈ C 1 (R n ) is bounded and that we have∑ n ∂ϕk=1 ∂x k(F )DF k ∈ L 2 (Π). In this case, obviously ϕ(F ) ∈ L 2 (F T ). We will employthe bump functions analyzed in Lemma 4.3.3. Let ε > 0 and define the sets K m andV m by K m = [−m, m] n and V m = (−m − 1, m + 1) n . By Lemma 4.3.3, there existsmappings (c m ) in Cc∞ (R n ) with K m ≺ c m ≺ V m such that the partial derivatives ofc m are bounded by 1 + ε.Define ϕ m by putting ϕ m (x) = c m (x)ϕ(x). We will argue that ϕ m ∈ C 1 (R n ) withbounded partial derivatives. It is clear that ϕ m is continuously differentiable, and∂ϕ m∂x k(x) = ∂c m∂x k(x)ϕ(x) + c m (x) ∂ϕ∂x k(x).

4.3 The Malliavin Derivative on D 1,2 147Since c m ∈ Cc∞ (R n ), both c m and ∂c m∂x khas compact support. It follows that ∂ϕ m∂x khas compact support, and since it is also continuous, it is bounded. We can thereforeapply the ordinary chain rule of Theorem 4.3.1 to conclude that ϕ m (F ) ∈ D 1,2and Dϕ m (F ) = ∑ n ∂ϕ mk=1 ∂x k(F )DF m . We will use the closedness of D to extend thisresult to ϕ(F ). We will show that ϕ m (F ) converges to ϕ(F ) in L 2 (F T ) and that∑ n ∂ϕ mk=1 ∂x k(F )DF k converges to ∑ n ∂ϕk=1 ∂x k(F )DF k in L 2 (Π). Closedness of D willthen yield ϕ(F ) ∈ D 1,2 and Dϕ(F ) = ∑ n ∂ϕk=1 ∂x k(F )DF k .Since c m converges pointwise to 1, ϕ m converges pointwise to ϕ. Therefore, we obtainϕ m (F ) a.s.−→ ϕ(F ). Because ‖c m ‖ ∞ ≤ 1, dominated convergence yields that ϕ m (F )converges in L 2 (F T ) to ϕ(F ). To show the corresponding result for the derivatives,first note that ∂c m∂x kis zero on Km. ◦ Since c m is one on Km, ◦ we find that ∂ϕ m∂x kare equal on Km. ◦ We therefore obtain==≤n∑∥k=1n∑∥k=1n∑∥k=1n∑∥k=1∂ϕ m∂x k(F )DF k −n∑k=1∥∂ϕ ∥∥∥∥(F )DF k∂x k2)∥ ∥∥∥∥DF k2(1 (K ◦ m ) c(F ) ∂cm(F )ϕ(F ) + (c m (F ) − 1) ∂ϕ ) ∥ ∥∥∥∥(F ) DF k∂x k ∂x k2∥ n∑1 (K ◦m ) c(F ) ( ∂ϕm∂x k(F ) − ∂ϕ∂x k(F )1 (K ◦ m ) c(F )∂c m∂x k(F )ϕ(F )DF k∥ ∥∥∥∥2+∥k=1and ∂ϕ∂x k∥1 (K ◦ m ) c(F )(c m(F ) − 1) ∂ϕ ∥∥∥∥(F )DF k .∂x k2We wish to show that each of these terms tend to zero by dominated convergence.Considering the first term, by definition of c m we obtain∣n∑∣k=11 (K ◦ m ) c(F )∂c m∂x k(F )ϕ(F )DF k∣ ∣∣∣∣≤ (1 + ε)‖ϕ‖ ∞n ∑k=1|DF k |,which is square-integrable. Likewise, for the second term we have the square-integrablebound∣ ∣ n∑1∣ (K ◦ m ) c(F )(c m(F ) − 1) ∂ϕ ∣∣∣∣ (F )DF k = |1 (K ◦∂x m ) c(F )(c n∑ ∂ϕ ∣∣∣∣m(F ) − 1)|(F )DFk ∣k∂x kk=1k=1∣ n∑ ∂ϕ ∣∣∣∣≤(F )DF k .∣ ∂x kSince K ◦ m increases to R n , by dominated convergence using the two bounds obtainedabove, we find that both norms tends to zero and therefore we may finally concludek=1

148 The Malliavin Calculusthat ∑ n ∂ϕ mk=1 ∂x k(F )DF k tends to ∑ nk=1implies that ϕ(F ) ∈ D 1,2 and Dϕ(F ) = ∑ nk=1∂ϕ∂x k(F )DF k in L 2 (Π). Closedness of D then∂ϕ∂x k(F )DF k .Step 2: The general case. Now let ϕ ∈ C 1 (R n ) be arbitrary and assume thatϕ(F ) ∈ L 2 (F T ) and ∑ n ∂ϕk=1 ∂x k(F )DF k ∈ L 2 (Π). We still consider some fixed ε > 0.Define the sets K m = [−m, m] and V m = [−(m + 1), m + 1] and let c m be the Lipschitzelement of Cc∞ (R) with K m ≺ c m ≺ V m that exists by Lemma 4.3.3 with Lipschitzconstant 1 + ε. Let C m be the antiderivative of c m which is zero at zero, given byC m (x) = ∫ x0 c m(y) dy. Note that since c m is bounded by one and is zero outside ofV m , ‖C m ‖ ∞ ≤ m + 1. In particular, C m is bounded. Defining ϕ m (x) = C m (ϕ(x)), itis then clear that ϕ m is bounded. Furthermore, ϕ m ∈ C 1 (R n ) andk=1∂ϕ m(x) = c m (ϕ(x)) ∂ϕ (x).∂x k ∂x kWe therefore have∣ ∣ ∣ n∑ ∂ϕ ∣∣∣∣ mn∑(F )DF k =∣ ∂x k ∣ c ∂ϕ ∣∣∣∣ n∑ ∂ϕ ∣∣∣∣m(ϕ(F )) (F )DF k ≤(F )DF k ,∂x k ∣ ∂x kk=1so ∑ n ∂ϕ mk=1 ∂x k(F )DF k ∈ L 2 (Π). Thus, ϕ m is covered by the previous step of the proof,and we may conclude that ϕ m (F ) ∈ D 1,2 and Dϕ m (F ) = ∑ n ∂ϕ mk=1 ∂x k(F )DF k . As inthe previous step, we will extend this result to ϕ by applying the closedness of D. Wewill therefore need to prove convergence in L 2 (F T ) of ϕ m (F ) to ϕ(F ) and to proveconvergence in L 2 (Π) of ∑ nk=1∂ϕ m∂x k(F )DF k to ∑ nk=1k=1∂ϕ∂x k(F )DF k .To this end, note that since c m is bounded by one, |C m (x)| ≤ ∫ x0 |c m(y)| dy ≤ |x|, andfor x ∈ [−m, m], C m (x) = ∫ x0 c m(y) dy = x. Therefore, ϕ m (x) = ϕ(x) on [−m, m] and‖ϕ m (F ) − ϕ(F )‖ 2 = ‖1 [−m,m] c(F )(ϕ m (F ) − ϕ(F ))‖ 2= ‖1 [−m,m] c(F )(C m (ϕ(F )) − ϕ(F ))‖ 2 .Since |C m (ϕ(F ))−ϕ(F )| ≤ |C m (ϕ(F ))|+|ϕ(F )| ≤ 2|ϕ(F )| by the contraction propertyof C m , we conclude by dominated convergence that the above tends to zero, so ϕ m (F )tends to ϕ(F ) in L 2 (F T ). Likewise, for the derivatives we obtain∥==n∑ ∂ϕ m(F )DF∥k −∂x kk=1n∑∥k=1n∑∥ (c m(ϕ(F )) − 1)n∑k=1c m (ϕ(F )) ∂ϕ∂x k(F )DF k −k=1∥∂ϕ ∥∥∥∥(F )DF k∂x k2n∑k=1∥∂ϕ ∥∥∥∥(F )DF k ,∂x k2∂ϕ∂x k(F )DF k∥ ∥∥∥∥2

4.3 The Malliavin Derivative on D 1,2 149and since c m −1 tends to zero, bounded by the constant one, we conclude by dominatedconvergence that the above tends to zero. By the closedness of D, we finally obtainϕ(F ) ∈ D 1,2 and Dϕ(F ) = ∑ n ∂ϕk=1 ∂x k(F )DF k , proving the theorem.Corollary 4.3.6. Let F, G ∈ D 1,2 . If the integrability conditions F G ∈ L 2 (F T ) and(DF )G + F (DG) ∈ L 2 (Π) hold, F G ∈ D 1,2 and D(F G) = (DF )G + F (DG).Proof. Putting ϕ(x, y) = xy, the chain rule of Theorem 4.3.5 yields that F G ∈ D 1,2and DF G = ∂ϕ∂x 1(F, G)DF + ∂ϕ∂x 2(F, G)DG = G(DF ) + F (DG).We are now done with the proof of the chain rule for the Malliavin derivative. Next,we prove that stochastic integrals with deterministic integrands are in the domain ofD. This will allow us to reconcile our definition of the Malliavin derivative with thatfound in Nualart (2006).Lemma 4.3.7. Let h ∈ L 2 [0, T ].D ∫ Th(t) dW (t) = h.0It then holds that ∫ T0 h(t) dW (t) ∈ D 1,2, andProof. First assume that h is continuous. We define h n (t) = ∑ nk=1 h( k n )1 ( k−1n , k ](t). nSince h is continuous, h is bounded and it follows that h n ∈ bE. We may thereforeconclude ∫ T0 h n(t) dW (t) ∈ S and∫ T∑n ( ( ( ( ))k k k − 1D s h n (t) dW (t) = D s h W − W0n)n)nk=1n∑( k= h 1n)(k−1n , n] (s) kk=1= h n (s).Next, note that h n converges pointwise to h for all t ∈ (0, T ]. By boundedness of h,h n converges in L 2 [0, T ] to h. By continuity of the stochastic integral, ∫ T0 h n(t) dW (t)converges in L 2 (F T ) to ∫ T0 h(t) dW (t). Closedness then yields ∫ T0 h(t) dW (t) ∈ D 1,2and D ∫ Th(t) dW (t) = h, as desired. This shows the result for continuous h.0Next, consider a general h ∈ L 2 [0, T ]. There exists continuous h n converging in L 2 [0, T ]to h. By the previous step, D ∫ T0 h n(t) dW t = h n . Thus, ∫ T0 h n(t) dW t converges to∫ T0h(t) dW t and D ∫ T0 h n(t) dW t converges to h. We conclude that ∫ T0 h(t) dW t ∈ D 1,2and D ∫ T0 h(t) dW t = h.

150 The Malliavin CalculusIn the following, we will for brevity denote the the Brownian stochastic integral operatoron [0, T ] by θ, that is, whenever X ∈ L 2 (Π) is progressively measurable, wewill write θ(X) = θX = ∫ T0 X s dW s . Furthermore, if X 1 , . . . , X n ∈ L 2 (Π) and we putX = (X 1 , . . . , X n ), we will write θX for (θX 1 , . . . , θX n ).Now consider h 1 , . . . , h n ∈ L 2 [0, T ] and let ϕ ∈ Cp ∞ (R n ). Since θh has a normaldistribution by Lemma 3.7.4, it follows that ϕ(θh) ∈ L 2 (Π). Furthermore, since h isdeterministic,∥ n∑ ∂ϕ ∥∥∥∥ n∑∥ (θh)h∥k ≤∂ϕ ∥∥∥2 n∑∂x k∥ (θh)h k =∂ϕ∂x k∥ (θh)∂x k∥ ‖h k ‖ 2,k=12 k=1k=12which is finite. By Lemma 4.3.7 and Theorem 4.3.5, we conclude that ϕ(θh) ∈ D 1,2and Dϕ(θh) = ∑ n ∂ϕk=1 ∂x k(θh)h k . This result shows that our method of introducingthe Malliavin derivative only for variables ϕ(W t ) with ϕ ∈ Cp ∞ (R n ) yields the sameresult as the method in Nualart (2006), where the Malliavin derivative is introducedfor variables ϕ(θh) with ϕ ∈ Cp ∞ (R n ). The two methods are thus equivalent.We let S h denote the space of variables ϕ(θh), where h = (h 1 , . . . , h n ), h k ∈ L 2 [0, T ]and ϕ ∈ C ∞ p (R n ). Clearly, S ⊆ S h . Our final goal of this section will be to prove afew versions of the integration-by-parts formula. This formula will be very useful inthe next section, where we develop the Hilbert space theory of the Malliavin calculus.If X ∈ L 2 (Π) and h ∈ L 2 [0, T ], we will write 〈X, h〉 [0,T ] (ω) = ∫ X(ω, t)h(t) dt.Theorem 4.3.8 (Integration-by-parts formula). Let h ∈ L 2 [0, T ] and F ∈ D 1,2 .Then E〈DF, h〉 [0,T ] = EF (θh), where F (θh)(ω) = F (ω)(θh)(ω).Proof. We first check that the conclusion is well-defined. Considering h as an elementof L 2 (Π), the left-hand side is simply the inner product in L 2 (Π) of two elements inL 2 (Π). And since F and θh are in L 2 (F T ), F (θh) ∈ L 1 (F T ). Thus, all the integralsare well-defined.By linearity, it will suffice to prove the result in the case where h has unit norm.First consider the case where F ∈ S h with F = f(θg), g ∈ (L 2 [0, T ]) n . By using alinear transformation, we can assume without loss of generality that g 1 = h and thatg 1 , . . . , g n are orthonormal. We then obtain〈∑ n∂fE〈DF, h〉 [0,T ] = E (θg)g k , g 1 = E∂x k〉[0,T ]k=1n∑k=1∂f∂x k(θg)〈g k , g 1 〉 [0,T ] = E ∂f∂x 1(θg).

4.3 The Malliavin Derivative on D 1,2 151Since g 1 , . . . , g n are orthonormal, θg is standard normally distributed. Therefore, wecan use Lemma 4.2.6 to obtainE ∂f ∫∫∂f(θg) = (y)φ n (y) dy = f(y)y 1 φ n (y) dy = Ef(θg)(θg) 1 = EF θ(h).∂x 1 ∂x 1This proves the claim in the case where F ∈ S h . Now consider the case where F ∈ D 1,2 .Let F n be a sequence in S h such that F n converges to F and DF n converges to DF , thisis possible by the closedness definition of D and our earlier observation that S ⊆ S h .By continuity of the inner products, we findas desired.E〈DF, h〉 [0,T ] = E〈lim DF n , h〉 [0,T ]= lim E〈DF n , h〉 [0,T ]= lim EF n θ(h)= E lim F n θ(h)= EF θ(h),Theorem 4.3.8 is the fundamental result known as the integration-by-parts formula,even though it may not seem to bear much similarity to the integration-by-parts formulaof ordinary calculus. In practice, we will not really use the integration-by-partsformula in the form given in Theorem 4.3.8, rather we will be using the following twocorollaries.Corollary 4.3.9. Let F, G ∈ D 1,2 with F G ∈ L 2 (F T ) and F (DG) + G(DF ) ∈ L 2 (Π).ThenE(G〈DF, h〉 [0,T ] ) = E (F G(θh)) − EF 〈DG, h〉 [0,T ]Comment 4.3.10 Note that the left-hand side of the formula above also can be writtenas ∫ ∫ DF (ω, t)G(ω)h(t) dt dP . This makes this formula useful for identifying[0,T ]conditional expectations, which in the Hilbert space context corresponds to orthogonalprojections.◦Proof. By Corollary 4.3.6, F G ∈ D 1,2 and D(F G) = F (DG) + G(DF ). Therefore,Theorem 4.3.8 yieldsE(F G(θh)) = E〈D(F G), h〉 [0,T ]= E〈F (DG), h〉 [0,T ] + E〈G(DF ), h〉 [0,T ]= EF 〈DG, h〉 [0,T ] + EG〈DF, h〉 [0,T ] ,

152 The Malliavin Calculusand rearranging yields the result.Corollary 4.3.11. Let F ∈ D 1,2 and G ∈ S h . ThenE(G〈DF, h〉 [0,T ] ) = E (F G(θh)) − EF 〈DG, h〉 [0,T ]Comment 4.3.12 The point of considering G ∈ S h is to avoid the integrabilityrequirements of Corollary 4.3.9. Particularly important for our later applications isthat we avoid the requirement F G ∈ L 2 (F T ).◦Proof. Let F n be a sequence in S h such that F n converges to F and DF n converges toDF . Then F n and G are both in S h , so F n and G are both Cp∞ transformations of coordinatesof the Wiener process. Since Cp∞ transformations and their partial derivativeshave polynomial growth, we conclude F n G ∈ L 2 (F T ) and F n (DG)+G(DF n ) ∈ L 2 (Π).From Lemma 4.3.9, we then haveE(G〈DF n , h〉 [0,T ] ) = E (F n G(θh)) − EF n 〈DG, h〉 [0,T ] .We want to show that each of these terms converge as F n converges to F .First term. We begin by considering the term on the left-hand side. Because wehave G ∈ L 2 (F T ) and h ∈ L 2 [0, T ], G ⊗ h ∈ L 2 (Π). DF n converges to DF in L 2 (Π),so by the Cauchy-Schwartz inequality, DF n (G ⊗ h) converges to DF (G ⊗ h) in L 1 (Π).Noting thatE ∣ ∣ ∫ T∫ T∣G〈DF n , h〉 [0,T ] − G〈DF, h〉 [0,T ] = E(G ⊗ h)DF∣n dλ − (G ⊗ h)DF dλ∣0≤ ‖DF n (G ⊗ h) − DF (G ⊗ h)‖ 1,we obtain that G〈DF n , h〉 [0,T ] converges in L 1 (F T ) to G〈DF, h〉 [0,T ] , and thereforeE(G〈DF n , h〉 [0,T ] ) converges to E(G〈DF, h〉 [0,T ] ), as desired.0Second term. Next, consider the first term on the right-hand side. Since G(θh) isin L 2 (F T ), the Cauchy-Schwartz inequality implies that F n G(θh) tends to F G(θh) inL 1 (F T ). Therefore E(F n G(θh)) converges to E(F G(θh)).Third term. Finally, consider the second term on the right-hand side. We proceedas for the term on the left-hand side and first note that since F n ∈ L 2 (F T ) andh ∈ L 2 [0, T ], F n ⊗ h ∈ L 2 (Π). We also obtain ‖F n ⊗ h − F ⊗ h‖ 2 2 = ‖F n − F ‖ 2 2‖h‖ 2 2,

4.3 The Malliavin Derivative on D 1,2 153showing that F n ⊗ h converges in L 2 (Π) to F ⊗ h. Therefore, (F n ⊗ h)DG convergesin L 1 (Π) to (F ⊗ h)DG. Since we have, as for the first term considered,E ∣ ∣ ∣F n 〈DG, h〉 [0,T ] − F 〈DG, h〉 [0,T ] = E∣∫ T0(F n ⊗ h)DG dλ −∫ T≤ ‖(F n ⊗ h)DG − (F ⊗ h)DG‖ 10(F ⊗ h)DG dλ∣we conclude that F n 〈DG, h〉 [0,T ] tends to F 〈DG, h〉 [0,T ] in L 1 (F T ), and thereforeE(F n 〈DG, h〉 [0,T ] ) tends to E(F 〈DG, h〉 [0,T ] ).Conclusion. We have now argued thatE(F Gθ(h)) = lim E(F n Gθ(h))nE(F 〈DG, h〉 [0,T ] ) = lim E(F n 〈DG, h〉 [0,T ] )nE(G〈DF, h〉 [0,T ] ) = lim E(G〈DF n , h〉 [0,T ] ),nand we may therefore conclude E(G〈DF, h〉 [0,T ] ) = E (F Gθ(h)) − EF 〈DG, h〉 [0,T ] bytaking the limits. This is the result we set out to prove.We have now come to the conclusion of this section. Before proceeding, we will reviewour progress. We have developed some of the fundamental results of the Malliavinderivative on D 1,2 . In Lemma 4.3.7, we have checked that stochastic integrals with deterministicintegrands are in D 1,2 . This allowed us to reconcile our Malliavin derivativewith that of Nualart (2006). Also, in Theorem 4.3.5 and Theorem 4.3.8, we provedthe chain rule and the integration-by-parts formula, respectively. These two resultswill be indispensable in the coming sections. The chain rule tells us that as long asnatural integrability conditions are satisfied, D 1,2 is stable under C 1 transformations.The integration-by-parts formula is essential because it can transfer an expectationinvolving DF into an expectation involving only F . This will be of particular use inthe form of Corollary 4.3.11.Our next goal is to introduce a series of subspaces of L 2 (F T ) and L 2 (Π) and investigatethe special properties of D related to these subspaces. These results are very importantfor the general theory of the Malliavin calculus. Unfortunately we will not be able tosee the full impact of them with the limited theory we detail here. However, we willtake the time to obtain one corollary from the theory, namely an extension of the chainrule to Lipschitz transformations.

154 The Malliavin Calculus4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π)In this section, we will introduce orthogonal decompositions of L 2 (F T ) and L 2 (Π)given by∞⊕∞⊕L 2 (F T ) = H n and L 2 (Π) = H n (Π).n=0We will see that subspaces H n have orthogonal bases for which the Malliavin derivativecan be explicitly computed, and the images of the subspaces are in H n (Π). Theseproperties will allow us a characterization of D 1,2 in terms of the projections on thesubspaces H n and H n (Π), which can lead to some very useful results. In order to givean idea of what is to come, we provide the following graphical overview.n=0H nspan H ′′ nH ′′ n(Π)spanH n (Π)H ′ nclcl H nD H n (Π) H ′ n(Π)clclConsider the left-hand part of graph. It is meant to state that H n is the closureof H n ′ and H n, ′′ and the span of H n is H n.′′ On the right-hand part, H n (Π) is theclosure of H n(Π) ′ and H n(Π), ′′ and the span of H n (Π) is H n. ′′ The connection betweenthe left-hand and right-hand parts is the Malliavin derivative, which maps H n intoH n (Π).Our plan is first to formally introduce H n , H ′ n, H ′′ n and H n and consider their basicproperties. After this, we introduce H n (Π), H ′ n(Π), H ′′ n(Π) and H n (Π). Finally, weprove that D maps H n into H n (Π) by explicitly calculating the Malliavin derivativeof the elements in H n . We then apply our results to obtain a characterization of theelements of the space D 1,2 in terms of the projections on the spaces H n .The subspaces H n are based on the Hermite polynomials. As described in AppendixA.5, the n’th Hermite polynomial is defined byH n (x) = (−1) n e x22d n x2e− 2dxn ,with H 0 (x) = 1 and H −1 (x) = 0. See the appendix for elementary properties of theHermite polynomials. This section will use a good deal of Hilbert space theory, seeAppendix A.6 for a review of the results used. We now define some of the subspacesof L 2 (F T ) under consideration. Recall that we in the previous section defined θ as theBrownian stochastic integral operator on [0, T ].

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 155Definition 4.4.1. We let H n′ be the linear span of H n (θh) for h ∈ L 2 [0, T ] with‖h‖ = 1 and let H n be the closure of H n ′ in L 2 (F T ).Our first goal is to show that the subspaces (H n ) are mutually orthogonal and thattheir orthogonal sum is L 2 (F T ). The following lemma will be fundamental to thisendeavor.Lemma 4.4.2. It holds that for h, g ∈ L 2 [0, T ] with ‖h‖ 2 = ‖g‖ 2 = 1,EH n (θh)H m (θg) ={0 if n ≠ mn!〈h, g〉 n if n = m .Proof. Let n, m ≥ 1 be given. By Lemma A.5.2, we have( )∣EH n (θh)H m (θg) = E ∂n∂s n exp sθh − s2 ∣∣∣s=0 ∂ m2( ( ) ∂n= E∂s n exp sθh − s2 ∂m2= E( ∂n∂ m (∂s n ∂t m exp sθh − s22∂t m exp (tθg − t2 2)∣∣∣∣t=0))∣ (tθg∂t m exp − t2 ∣∣∣s=0,t=02) ( ))∣exp tθg − t2 ∣∣∣s=0,t=0.2Our plan is to interchange the differentiation and integration above and explicitlycalculate and differentiate the resulting mean.Step 1: Interchange of differentiation and integration. We first note that by asimple induction proof,∂ n ∂ m ( ) ( )) ( )∂s n ∂t m exp sx − s2exp ty − t2 = q n (s, x)q m (t, y) exp(sx − s2exp ty − t2 ,2222where q n is some polynomial in two variables. We will show that for any s, t ∈ R, itholds thatE ∂ ) ( )∂s q n(s, x)q m (t, y) exp(sθh − s2exp tθg − t222= ∂ ) ( )∂s Eq n(s, x)q m (t, y) exp(sθh − s2exp tθg − t2 .22By symmetry, if this is true, then the same result will of course also hold when differentiationwith respect to t. Applying this result n + m times will yield the desiredexchange of differentiation and integration. To show the result, fix t ∈ R and consider

156 The Malliavin Calculusa bounded interval [a, b]. Define c = min{|a|, |b|}. We then have, with r n+1 somepolynomial and C and C t ′ some suitably large constants,) ( )∣ sup∂∣a≤s≤b ∂s q n(s, x)q m (t, y) exp(sx − s2exp ty − t2 ∣∣∣22 ) ( )∣ = sup∣ q n+1(s, x)q m (t, y) exp(sx − s2exp ty − t2 ∣∣∣a≤s≤b22( ) ( )≤ r n+1 (|x|)q m (|t|, |y|) exp bx − c2 exp ty − t222( ) ( )≤ (C + exp(|x|)(C t ′ + exp(|y|)) exp bx − c2 exp ty − t2 .22We have exploited that exponentials grow faster than any polynomial. Now, sinceE exp( ∑ nk=1 λ k|X k |) is finite whenever λ ∈ R n and X has an n-dimensional normaldistribution, we find( ) ( )E(C + exp(|θh|)(C t ′ + exp(|θg|)) exp bθh − c2 exp tθg − t2 < ∞.22Ordinary results for exchanging differentiation and integration now certify thatE ∂ ) ( )∂s q n(s, x)q m (t, y) exp(sθh − s2exp tθg − t222= ∂ ) ( )∂s Eq n(s, x)q m (t, y) exp(sθh − s2exp tθg − t2 ,22as desired. We can then concludeE( ∂n∂ m (∂s n ∂t m exp sθh − s22= ∂n∂s n ∂ m∂t m E (exp(sθh − s22) ( ))exp tθg − t2 2) ( ))exp tθg − t2 ,2and finally,( ∂n∂ m (E∂s n ∂t m exp sθh − s22= ∂n ∂ m (∂s n ∂t m E exp sθh − s22) ( ))∣exp tθg − t2 ∣∣∣s=0,t=02)exp(tθg − t2 2)∣∣∣∣s=0,t=0.Step 2: Calculating the mean. We now set to work to calculate) ( )E exp(sθh − s2exp tθh − t2 .22

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 157By Lemma 3.7.4, (θh, θg) is normally distributed with mean zero, V θh = 1, V θg = 1and Cov(θh, θg) = 〈h, g〉. We then conclude for any s, t ∈ R, sθh + tθg is normallydistributed with mean zero and variance s 2 + t 2 + 2st〈h, g〉. Using the formula for theLaplace transform of the normal distribution, we then findE exp(sθh − s22= exp(− s2 + t 22= exp(− s2 + t 22= exp (st〈h, g〉) .) ( )exp tθg − t2 2)E exp (sθh + tθg)) ( )1exp2 (s2 + t 2 + 2st〈h, g〉)Step 3: Conclusion. With α = max{n, m}, we now obtain∂ n ∂ m ( ) ( )∂s n ∂t m E exp sθh − s2exp tθg − t222= ∂n ∂ m∂s n exp (st〈h, g〉)∂tm = ∂n ∂ m ∑∞ 〈h, g〉 k∂s n ∂t ms k t kk!k=0∞∑ 〈h, g〉 k=n!m!s k−n t k−m .k!k=αCombining our results, we conclude( ∂n∂ m (EH n (θh)H m (θg) = E∂s n ∂t m exp sθh − s22= ∂n ∂ m (∂s n ∂t m E exp sθh − s22∞∑=k=α) ( ))∣exp tθg − t2 ∣∣∣s=t=02)exp∣〈h, g〉 k∣∣∣∣n!m!s k−n t k−m ,k!s=t=0( )∣tθg − t2 ∣∣∣s=0,t=02and the final expression above is zero if n ≠ m and n!〈h, g〉 n otherwise. This showsthe claim of the lemma.Theorem 4.4.3. The subspaces (H n ) are orthogonal, and L 2 (F T ) = ⊕ ∞ n=0H n .Proof. We first prove that the subspaces are mutually orthogonal. Let n, m ≥ 1 begiven with n ≠ m and let h, g ∈ L 2 [0, T ] with ‖h‖ 2 = ‖g‖ 2 = 1. By Lemma 4.4.2,

158 The Malliavin CalculusH n (θh) and H m (θg) are orthogonal. Since H n is the closure of the span of elementsof the form H n (θh), ‖h‖ = 1, Lemma A.6.2 yields that H n ⊥H m .Next, we must show L 2 (F T ) = ⊕ ∞ n=0H n . By Lemma A.6.11, ⊕ ∞ n=0H n is the closureof span ∪ ∞ n=0 H n . It will therefore suffice to show that the subspace span ∪ ∞ n=0 H n isdense in L 2 (F T ). To do so, it will by Lemma A.6.4 suffice to show that the orthogonalcomplement of span ∪ ∞ n=0 H n is {0}. And to do so, it will by Lemma A.6.2 suffice toshow that the orthogonal complement of ∪ ∞ n=0H n is {0}. This is therefore what weset out to do.Therefore, let F ∈ L 2 (F T ) be given, orthogonal to ∪ ∞ n=1H n . This means that, for anyn ≥ 0 and h with ‖h‖ = 1, EF H n (θh) = 0. Let n be given. By Lemma A.5.6, thereare coefficients such thatn∑x n = λ k H k (x).k=0We then obtain EF (θh) n = ∑ nk=0 λ kEF H k (θh) = 0. By scaling, we can conclude thatEF (θh) n for any h ∈ L 2 [0, T ]. Therefore, in the following h will denote an arbitraryelement of L 2 [0, T ], not necessarily of unit norm. We will show EF exp(θh) = 0. Todo so, note that by Lemma B.3.2,∞∑(θh) n∥ n! ∥ =2n=0==≤∞∑n=0∞∑n=0∞∑n=0∞∑n=0∑ ∞n=01 (E(θh)2n ) 1 2n!(1 (2n)!n! ‖h‖n 2 n n!) 12‖h‖ n ( ) 1(2n)!2√n! 2 n (n!) 2‖h‖ n√n!,where we have used that (2n)!2 n n!≤ n!.is convergent by the quotient test,and by completeness, we can conclude that ∑ ∞ (θh) nn=0 n!is convergent in L 2 (F T ). Sinceit also converges almost surely to exp(θh), this is also the limit in L 2 (F T ). Bythe Cauchy-Schwartz inequality, we find that F ∑ n (θh) kk=0 k!converges in L 1 (F T ) toF exp(θh), and thereforeEF exp(θh) = limnEF‖h‖ n√n!n∑ (θh) k= 0.k!Since the variables of the form exp(θh) where h ∈ L 2 [0, T ] form a dense subset ofk=0

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 159L 2 (F T ) by Theorem B.6.2, their span form a dense subspace of L 2 (F T ). Since F isorthogonal to this subspace, Lemma A.6.4 yields F = 0, as desired.Next, we introduce the orthonormal basis H n for the subspace H n . We are going toneed an orthonormal basis for L 2 [0, T ]. Since B[0, T ] is countably generated, LemmaB.4.1 yields that L 2 [0, T ] is separable, and therefore any orthonormal basis must becountable. Let (e n ) be such a basis, we will hold this basis fixed for the remainder ofthe section.We let I be the set of sequences with values in N 0 which are zero from a point onwards.We call the elements of I multi-indices. If a ∈ I, we put |a| = ∑ ∞n=1 a n and call |a| thedegree of the multi-index. We also define a! = ∏ ∞n=1 a n!. We let I n be the multi-indicesof degree n and let I m n be the multi-indices a of degree n such that a k = 0 wheneverk > m. We say that the multi-indices in I m n have order m.Definition 4.4.4. For any multi-index a of order n, we define Φ a = 1 √a!∏ ∞n=1 H a n(θe n ).We write H = {Φ a |a ∈ I} and H n = {Φ a ||a| = n}. We let H ′′ n denote the span of H n .We begin by showing that H is an orthonormal set.showing that H n is an orthonormal basis for H n .Afterwards, we will work onLemma 4.4.5. H is an orthonormal set in L 2 (F T ).Proof. We need to show that each Φ a has unit norm and that the elements of thefamily is mutually orthogonal.Unit norm. Let a multi-index a be given. Since (e n ) is orthonormal, the family(θe n ) are mutually independent. Recalling that a k is zero from a point onwards, usingLemma 3.7.4 and Lemma 4.4.2, we obtain(‖Φ a ‖ 2 1 ∏∞2 = E √ H an (θe n )a!= 1 a!= 1 a!= 1.n=1∞∏EH an (θe n ) 2n=1∞∏a n !‖e n ‖ 2n=1) 2

160 The Malliavin CalculusOrthogonality. Next, consider two multi-indices a and b with a ≠ b. We then find() ()1 ∏∞ 1 ∏∞〈Φ a , Φ b 〉 = E √ H an (θe n ) √ H bn (θe n )a! b!==1√a!√b!En=1n=1∞∏H an (θe n )H bn (θe n )n=11 ∏ ∞√ √a! b!n=1EH an (θe n )H bn (θe n ).Now, since there is some n such that a n ≠ b n , it follows from Lemma 4.4.2 that forthis n, EH an (θe n )H bn (θe n ) = 0. Therefore, the above is zero and Φ a and Φ b areorthogonal.In order to show that H n is an orthonormal basis for H n , we will need to add a few morespaces to our growing collection of subspaces of L 2 (F T ). Recall that P n denotes thepolynomials of degree less than or equal to n. That is, P n is the family of polynomialsof degree less than or equal to n in k variables for any k ≥ 1.Definition 4.4.6. We let P ′ n be the linear span of p(θh), where p ∈ P n is a polynomialin k variables and h ∈ (L 2 [0, T ]) k , and let P n be the closure of P ′ n.Lemma 4.4.7. Consider h ∈ (L 2 [0, T ]) n with h = (h 1 , . . . , h n ). Let (h k ) be a sequencewith h k = (h k 1, . . . , h k n) converging to h. Let f ∈ C 1 p(R n ). Then f(θh k ) converges tof(θh) in L 2 (F T ).Proof. This follows from Lemma B.3.4.Lemma 4.4.8. Let P n be the family of variables p(θe), where p ∈ P n is a polynomiumof k variables and e is a vector of distinct elements from the orthonormal basis (e n ).Then the closure of the span of P n is equal to P n .Proof. Obviously, P n ⊆ P n, ′ and therefore span P n ⊆ P n . We need to prove the otherinclusion. We first consider the case F = p(θh), where p ∈ P n is a polynomial of degreen in k variables and h = (h 1 , . . . , h k ) with coordinates in span {e i } i≥1 . We want toprove that F ∈ span P n . To do so, first observe that there exists an orthonormalsubset {g 1 , . . . , g m } of distinct elements from {e i } i≥1 such that h j = ∑ mi=1 λj i g i forsome coefficients λ j i . We then obtain( m)∑m∑p(θh) = p λ 1 i g i , . . . , λ k i g i = q(g 1 , . . . , g m ),i=1i=1

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 161where q(x 1 , . . . , x m ) = p( ∑ mi=1 λ1 i x i, . . . , ∑ mi=1 λk i x i) is a polynomium of degree lessthan or equal to n in m variables. We conclude p(θh) ∈ P n . We have now proven thatp(θh) is in span P n when the coordinates of h are in span {e i } i≥1 .Next, consider F = p(θh), where h ∈ (L 2 [0, T ]) k , h = (h 1 , . . . , h k ). We need to provethat p(θh) is in span P n . To this end, note that since (e n ) is an orthonormal basis, thereexists sequences (h i j ) in span {e i} i≥1 for j ≤ k such that h i j converges to h j. Puttingh i = (h i 1, . . . , h i k ), we then find by Lemma 4.4.7 that p(θhi ) converges in L 2 to p(θh).Since we have already proven that p(θh i ) ∈ span P n , this shows that p(θh) ∈ span P n .By linearity, we may now conclude P ′ n ⊆ span P n and therefore also P n ⊆ span P n , asdesired.Lemma 4.4.9. For each n ≥ 0, P n = ⊕ n k=0 H k.Proof. We first show the easier inclusion ⊕ n k=0 H k ⊆ P n . Let k ≤ n. By Lemma A.5.6,the k’th Hermite polynomial is a polynomial of degree k, therefore an element of P n .Thus, H ′ k ⊆ P′ n ⊆ P n . Since P n is closed, this implies H k ⊆ P n , and we thereforeimmediately obtain ⊕ n k=0 H k ⊆ P n , as desired.Now consider the other inclusion, P n ⊆ ⊕ n k=0 H k. From Theorem 4.4.3 and LemmaA.6.16, we have the orthogonal decomposition(∞⊕n⊕) ( ∞)⊕L 2 (F T ) = H k = H k ⊕ H k .k=0k=0k=n+1By Lemma A.6.14, we therefore find that to show P n ⊆ ⊕ ∞ k=0 H k, it will suffice toshow that P n is orthogonal to ⊕ ∞ k=n+1 H k. And to do so, it will by Lemma A.6.11 andLemma A.6.2 suffice to show that P n is orthogonal to Hk ′ for any k > n.Therefore, let k > n be given, and let ‖h‖ = 1. We will show that P n is orthogonalto H k (θh). To this end, it will suffice to show that P n ′ is orthogonal to H k (θh).Let F ∈ P n ′ with F = p(θg), where g = (g 1 , . . . , g m ) are in L 2 [0, T ]. Using thesame technique as in the proof of Lemma 4.4.8, we can assume that g 1 , . . . , g m areorthonormal. In fact, by the Gram-Schmidt orthonormalization procedure, we caneven assume g 1 = h.We now prove EF H k (θh) = 0. By Lemma A.5.6, we have, for any i ≥ 0,i∑x i = µ i jH j (x)j=0

162 The Malliavin Calculusfor some coefficients µ i j ∈ R. Recalling that Im n denotes the multi-indicies of order mwith degree n, we then have, for some coefficients λ a ,Therefore, we obtainp(θg) = ∑ ∏ mλ a (θg i ) a ia∈I m na∈I m ni=1= ∑ ∏m ∑a iλ a µ a ij H j(θg i )= ∑EF H k (θh) = ∑a∈I m na∈I m n∑a 1λ ai=1 j=0∑a 1λ aj 1 =0j 1 =0· · ·· · ·a m∑j m =0a m∑m∏j m =0 i=1EH k (θh)µ a ij iH ji (θg i ).m∏i=1µ a ij iH ji (θg i ).Now consider the expression EH k (θh) ∏ mi=1 µai j iH ji (θg i ), we wish to argue that this isequal to zero. Since g 1 , . . . , g m are orthonormal, θg 1 , . . . , θg m are independent. Wetherefore find, recalling that g 1 = h,EH k (θh)m∏i=1µ a ij iH ji (θg i ) = E(H k (θh)H j1 (θh))m∏i=2Eµ a ij iH ji (θg i ).Since |a| ≤ n, a 1 ≤ n and in particular j 1 ≤ n. Since k > n, we may conclude byLemma 4.4.2 that E(H k (θh)H j1 (θh)) = 0, and the above expression is therefore zero.Thus, EF H k (θh) = 0 and we conclude that P n is orthogonal to Hk ′ for k > n. As aconsequence of this, P n is orthogonal to ⊕ ∞ k=n+1 H k and therefore P n ⊆ ⊕ n k=0 H k, asdesired.Theorem 4.4.10. For each n, H n is an orthonormal basis of H n .Proof. From Lemma 4.4.5, we already know that H n is an orthonormal set, so it willsuffice to show that the span H ′′ n is dense in H n . Let K n be the closure of H ′′ n. Wewish to show K n = H n . To do so, we first prove P n = ⊕ n k=0 K k. Since Φ a for |a| = kis a polynomial transformation of degree k of variables θ(e i ), it is clear that H k ⊆ P nfor any k ≤ n and therefore K k ⊆ P n . In particular, ⊕ n k=0 K k ⊆ P n . We need to provethe other inclusion.By Lemma 4.4.8, it will suffice to show P n ⊆ ⊕ n k=0 K k. Therefore, let F ∈ P nwith F = p(θe), where e = (e k1 , . . . , e kn ) are distinct and p(x) = ∑ a∈I λ kx a is amn

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 163polynomial of degree n in m variables. By Lemma A.5.6, we have, for any n ≥ 0,x n = ∑ nk=0 µn k H k(x), for some coefficients µ n k∈ R. We obtainp(θ(e k1 ), . . . , θ(e kn )) = ∑a∈I m n= ∑a∈I m n= ∑a∈I m n= ∑λ aλ an∏i=1(θe ki ) a i∏n ∑a ii=1 j=0∑a 1λ a∑a 1a∈I m n j 1 =0j 1 =0· · ·· · ·µ a ij H j(θe ki )a n∑n∏j n =0 i=1a n∑j n =0(∏nλ ai=1µ a ij iH ji (θe ki )√ji !µ aij i) n∏i=11√ji ! H j i(θe ki ).Now, with |j| = ∑ nk=1 j k, the innermost product of the above is in H |j| . Since |a| ≤ nand j i ≤ a i for i ≤ n, |j| ≤ n. We may therefore conclude that the above is in thespan of ∪ n k=0 H k, therefore in ⊕ n k=0 K k.We have now shown P n = ⊕ n k=0 K k. Recall that from Lemma 4.4.9, we also haveP n = ⊕ n k=0 H k. We may then concludeP n−1 ⊕ K n =n⊕K k =k=0n⊕H k = P n−1 ⊕ H n .Now, since P n−1 = ⊕ n−1k=0 K k and P n−1 = ⊕ n−1k=0 H k, we conclude that K n and P n−1 areorthogonal and P n−1 and H n are orthogonal. Therefore, by Lemma A.6.15, K n = H nand therefore span H n = cl H ′′ n = K n = H n , as desired.k=0Our results so far are the following: The subspaces H n are mutually orthogonal, hasorthonormal bases H n , and L 2 (F T ) = ⊕ ∞ n=0H n . We also have two subspaces H ′ n andH ′′ n which are dense in H n .We will now use the families H n , H ′ n, H ′′ n and H n to define analogous families H n (Π),H ′ n(Π), H ′′ n(Π) and H n (Π) in L 2 (Π). These sets will have properties much alike to theircounterparts in L 2 (F T ), and will eventually be linked to these through the Malliavinderivative.Definition 4.4.11. By H ′ n(Π), we denote the span of the variables of the form F ⊗ h,where F ∈ H n and h ∈ L 2 [0, T ]. By H n (Π), we denote the closure of H ′ n(Π).

164 The Malliavin CalculusTheorem 4.4.12. Defining H n (Π) as the family of elements Φ a ⊗ e i , where |a| = nand i ∈ N, H n (Π) is an orthonormal basis for H n (Π). In particular, the span H ′′ n(Π)of H n (Π) is dense in H n (Π).Proof. It is clear that ‖Φ a ⊗ e i ‖ 2 = ‖Φ a ‖ 2 ‖e i ‖ 2 = 1. For any a, b ∈ I n and i, j ∈ N,the Fubini Theorem yields 〈Φ a ⊗ e i , Φ b ⊗ e j 〉 = 〈Φ a , Φ b 〉〈e i , e j 〉, so if a ≠ b or i ≠ j,Φ a ⊗ e i and Φ b ⊗ e j are orthogonal. Thus, H n (Π) is orthonormal.It remains to show span H n (Π) = H n (Π). To this end, it will suffice to show thatH ′ n(Π) ⊆ span H n (Π). Therefore, consider F ∈ H n and h ∈ L 2 [0, T ]. We need todemonstrate F ⊗ h ∈ span H n (Π). Now, there are h n ∈ span {e i } i≥1 such that h ntends to h, and thereforelimn‖F ⊗ h − F ⊗ h n ‖ 2 = limn‖F ‖ 2 ‖h − h n ‖ 2 = 0.Furthermore, there are F n ∈ span H n such that F n tends to F . Fixing n, we then findlimk‖F ⊗ h n − F k ⊗ h n ‖ 2 = limk‖F − F k ‖ 2 ‖h n ‖ = 0.Since F k ∈ span H n and h n ∈ span {e i } i≥1 , we obtain F k ⊗ H n ∈ span H n (Π). Now,since F k ⊗ h n tends to F ⊗ h n as k tends to infinity, F ⊗ h n ∈ span H n (Π). And sinceF ⊗ H n tends to F ⊗ h as n tends to infinity, F ⊗ h ∈ span H n (Π). This shows thestatement of the theorem.Theorem 4.4.13. The spaces H n (Π) are orthogonal and L 2 (Π) = ⊕ ∞ n=0H n (Π).Proof. Since the sets H n are orthogonal, it is clear that the sets H n (Π) are orthogonal.Therefore, H n (Π) are orthogonal as well.In order to show that L 2 (Π) = ⊕ ∞ n=0H n (Π), as in Theorem 4.4.3 it will suffice to showthat the orthogonal complement of ∪ ∞ n=0H n (Π) is zero. Therefore, let X ∈ L 2 (Π), andassume that X is orthogonal to H n (Π) for any n ≥ 0. In particular, X is orthogonalto F ⊗ h for any F ∈ H n and any h ∈ L 2 [0, T ]. Since L 2 (F T ) = ⊕ ∞ k=0 H k, we concludeby continuity of the inner product that X is orthogonal to F ⊗ h for any F ∈ L 2 (F T )and any h ∈ L 2 [0, T ]. By Lemma B.4.2, X is zero almost surely.Next, we will investigate the interplay between H n , H n (Π) and D. Here, our orthonormalbases H n will prove essential.

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 165Theorem 4.4.14. H ⊆ D 1,2 , and for any multi-index a,DΦ a =∞∑αke a k ,k=1where αk a = √ a ka!H ak −1(θe k ) ∏ ∞i≠k H a i(θe i ), with the convention that H −1 = 0. Theelements (DΦ a ) are mutually orthogonal, and ‖DΦ a ‖ 2 = |a|.Comment 4.4.15 Since a k is zero from some point onwards, the infinite product inαk a only has finitely many nontrivial factors, and is therefore well-defined. Similarly,we see that αk a is zero from some point onwards, and therefore the infinite sum in theexpression for DΦ a is well-defined, only having finitely many nontrivial terms. ◦Proof. We need to prove four things: That Φ a ∈ D 1,2 , the form of DΦ a , orthogonalityand the norm of ‖DΦ a ‖.Computation of DΦ a . Let a ∈ I and assume that a k = 0 for k > n. ThenΦ a = 1 √a!∏ nk=1 H a n(θe n ), so Φ a = f(H a1 (θe 1 ), . . . , H an (θe n )), where f : R n → Rwith f(x) = 1 √a!∏ nk=1 x k. By the chain rule, Φ a ∈ D 1,2 andDΦ a =====n∑ ∂f(H a1 (θe 1 ), . . . , H an (θe n )) DH ak (θe k )∂x k⎛⎞n∑⎝√ 1 ∏n H ai (θe i ) ⎠ a k H ak −1(θe k )e ka!k=1k=1n∑k=1i≠k⎛a√ k⎝H ak −1(θe k )a!n∑αke a kk=1∞∑αke a k ,k=1⎞n∏H ai (θe i ) ⎠ e kwhere we have used Lemma A.5.3. Note that in the case where a k = 0, H ak = 1, soH a ′ k= 0 = H −1 . The above computation is therfore valid no matter whether a k iszero or nonzero.Orthogonality. Let a and b be multi-indices. Recall that the inner product onL 2 (Π) is denoted 〈·, ·〉 Π . We have, using Lemma 3.7.4 and recalling that in the sumsi≠k

166 The Malliavin Calculusand products, only finitely many terms and factors are nontrivial,〈Φ a , Φ b 〉 Π = E〈Φ a , Φ b 〉 [0,T ]〈∑ ∞ ∞∑〉= E αke a k , αke b k===k=1∞∑Eαkα a kbk=1k=1[0,T ]⎛⎞ ⎛⎞∞∑ a k b√ √ k∞∏∞∏E ⎝H ak −1(θe k ) H ai (θe i ) ⎠ ⎝H bk −1(θe k ) H bi (θe i ) ⎠a! b!k=1i≠k⎛⎞∞∑ a k b√ √ k∞∏(EH ak −1(θe k )H bk −1(θe k )) ⎝ EH ai (θe i )H bi (θe i ) ⎠ .a! b!k=1In particular, if a ≠ b, 〈Φ a , Φ b 〉 Π = 0 by Lemma 4.4.2, showing orthogonality of distinctΦ a and Φ b .i≠ki≠kNorm of DΦ a . Finally, we find‖Φ a ‖ 2 Π ===∞∑ a 2 ka! EH ∏∞a k −1(θe k ) 2 EH ai (θe i ) 2k=1∞∑k=1∞∑k=1= |a|.a 2 ka! (a k − 1)!a 2 k (a k − 1)!a k !∞∏a i !i≠ki≠kTheorem 4.4.14 has several corollaries which gives us a good deal of insight into thenature of the Malliavin derivative.Corollary 4.4.16. It holds that H n ⊆ D 1,2 , and for any F ∈ H n , ‖DF ‖ = √ n‖F ‖.Proof. First consider F ∈ H ′′ n with F = ∑ mk=1 λ kΦ ak , where a k ∈ I n for k ≤ m. ByLemma 4.4.14 and the Pythagoras Theorem, we obtain F ∈ D 1,2 andm∑m∑m∑m∑‖DF ‖ 2 = λ 2 k‖DΦ ak ‖ 2 = λ 2 k|a k | = n λ 2 k = n λ 2 k‖Φ ak ‖ 2 = n‖F ‖ 2 .k=1k=1k=1k=1

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 167We have now shown that H ′′ n ⊆ D 1,2 and that for F ∈ H ′′ n, ‖DF ‖ = √ n‖F ‖. We needto extend this to H n . Therefore, let F ∈ H n be given, and let F n be a sequence in H ′′ nconverging to F . By what we already have shown, ‖DF n − DF m ‖ = √ n‖F n − F m ‖.Since (F n ) is a cauchy sequence, we conclude that DF n is a cauchy sequence as well.By completeness, there exists a limit X of DF n . By the closedness of D, we concludethat F ∈ D 1,2 and DF = X. We then have‖DF ‖ = ‖ lim DF n ‖ = lim ‖DF n ‖ = √ n lim ‖F n ‖ = √ n‖F ‖.This shows the claims of the corollary.Corollary 4.4.17. D maps H n into H n−1 (Π).Proof. Let Φ a ∈ H n be given. By Lemma 4.4.14,DΦ a =∞∑αke a k ,k=1where αk a = √ a ka!H ak −1(θe k ) ∏ ∞i≠k H a i(θe i ). By inspection, whenever αk a ≠ 0 we haveαk ae k ∈ H n−1(Π) ′′ ⊆ H n−1 (Π). By Corollary 4.4.16, D is continuous on H n . SinceH n−1 (Π) is a closed linear space, the result therefore extends from H n to H n .Corollary 4.4.18. The images of H n and H m under D are orthogonal whenevern ≠ m.Proof. This follows by combining Corollary 4.4.17 and Theorem 4.4.13.We have now introduced all of the subspaces and orthogonal sets described in thediagram in the beginning of the section, and we have discussed their basic properties.We are now ready to begin work on a characterisation of D 1,2 in terms of the projectionson the subspaces H n . Let P n be the orthogonal projection in L 2 (F T ) on H n , and letPnΠ be the orthogonal projection in L 2 (Π) on H n (Π).Lemma 4.4.19. If F ∈ H ′′ n and h ∈ L 2 [0, T ], F (θh) ∈ H n−1 ⊕ H n+1 .Proof. We prove the result by first considering some simpler cases and then extendingusing density arguments.

168 The Malliavin CalculusStep 1: The case F ∈ H n and h = e p . First consider the case where F ∈ H n withF = Φ a and h = e p for some p ∈ N. Supposing that a k = 0 for k > m, we can writeΦ a = √ 1 ∏ ma! i=1 H a i(θe i ). In the case p ≤ m, we find, using Lemma A.5.4,Φ a θ(e p ) ===1√a!H ap (θe p )θ(e p ) ∏ i≠pH ai (θe i )1 (√ Hap+1(θe p ) + a p H ap−1(θe p ) ) ∏H ai (θe i )a!1√ H ap+1(θe p ) ∏ H ai (θe i ) + √ 1 a p H ap−1(θe p ) ∏ H ai (θe i ),a!i≠pa!i≠pi≠pwhich is in H n−1 ⊕ H n+1 . If p > m, we obtain Φ a θ(e p ) = 1 √a!H 1 (θe p ) ∏ mi=1 H a i(θe i ),yielding Φ a θ(e p ) ∈ H n+1 ⊆ H n−1 ⊕ H n+1 .Step 2: The case F ∈ H n and h ∈ L 2 [0, T ]. By linearity, it is clear that the resultextends to F = Φ a and h in the span of {e i } i≥1 . Consider a general h ∈ L 2 [0, T ], andlet h n be in the span of {e i } i≥1 , converging to h. Now, the mapping()1 ∏mf(x) = √ H ai (x i )a!i=1x m+1is in C ∞ p (R m+1 ), and furthermore we have the equalities Φ a θ(h) = f(θe 1 , . . . , θe m , θh)and Φ a θ(h n ) = f(θe 1 , . . . , θe m , θh n ). Therefore, Lemma 4.4.7 yields that Φ a θ(h n )converges in L 2 (F T ) to Φ a θ(h). Since H n−1 ⊕ H n+1 is a closed subspace, we mayconclude Φ a θ(h) ∈ H n−1 ⊕ H n+1 .Step 3: The case F ∈ H ′′ n and h ∈ L 2 [0, T ]. The remaining extension followsdirectly by linearity as in the previous step.Lemma 4.4.20. If X ∈ H n (Π) and h ∈ L 2 [0, T ], 〈X, h〉 [0,T ] ∈ H n .Proof. Let h ∈ L 2 [0, T ]. First note that for any F ∈ H n and h ′ ∈ L 2 [0, T ], we find〈F ⊗ h ′ , h〉 [0,T ] = F 〈h ′ , h〉 ∈ H n , showing the result in this simple case. Since themapping X ↦→ 〈X, h〉 [0,T ] is linear and continuous and H n is a closed subspace, theresult extends to X ∈ H n (Π).Theorem 4.4.21. Let F ∈ D 1,2 . Then P Π n DF = DP n+1 F .Proof. Our plan is to show that 〈Z, DF 〉 = 〈Z, DP n+1 F 〉 for any Z ∈ H n (Π) and thenuse Lemma A.6.18 to obtain the conclusion. Showing this equality in full generality

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 169will be done in two steps, first demonstrating it in a simple case and then extendingby linearity and continuity.Step 1: The equality for G ⊗ h. Let G ∈ H ′′ n and let h ∈ L 2 [0, T ]. We will showthat 〈G ⊗ h, DF 〉 = 〈G ⊗ h, DP n+1 F 〉. Since H ′′ n ⊆ S h , we can use Corollary 4.3.11and obtain〈G ⊗ h, DF 〉 = E(G〈DF, h〉 [0,T ] )= EF G(θh) − EF 〈DG, h〉 [0,T ] .Using the orthogonal decomposition of L 2 (F T ) proven in Theorem 4.4.3, LemmaA.6.12 yields F = ∑ ∞k=0 P kF , where the limit is in L 2 (F T ). Using the Cauchy-Schwartz inequality, we then obtain the two results∞∑‖G(θh)P k F ‖ 1 ≤ ‖G(θh)‖ 2k=0∞ ∑k=0‖P k F ‖ 2 < ∞∞∑‖〈DG, h〉 [0,T ] P k F ‖ 1 ≤ ‖〈DG, h〉 [0,T ] ‖ 2k=0∞ ∑k=0‖P k F ‖ 2 < ∞,so ∑ ∞k=0 G(θh)P kF and ∑ ∞k=0 〈DG, h〉 [0,T ]P k F are convergent in L 1 . Since convergencein L p implies convergence in probability, we conclude by uniqueness of limits that∞∑∞∑G(θh)P k F = G(θh) P k Fk=0k=0∞∑∑∞〈DG, h〉 [0,T ] P k F = 〈DG, h〉 [0,T ] P k F,k=0where the limits on the left are in L 1 (F T ) and the limits on the right are in L 2 (F T ).This impliesk=0E(G〈DF, h〉 [0,T ] ) = EF G(θh) − EF 〈DG, h〉 [0,T ]( ∞) (∑∞)∑= E G(θh)P k F − E 〈DG, h〉 [0,T ] P k F=k=0k=0k=0∞∑∞∑EG(θh)P k F − E〈DG, h〉 [0,T ] P k F,where we have used the L 1 -convergence to interchange sum and integral. By assumption,G ∈ H ′′ n, so DG ∈ H n−1 (Π) by Corollary 4.4.17, and 〈DG, h〉 [0,T ] ∈ H n−1 byLemma 4.4.20. Furthermore, by Lemma 4.4.19, Gθ(h) ∈ H n−1 ⊕ H n+1 . By orthogo-k=0

170 The Malliavin Calculusnality of the H n , we then find∞∑∞∑EG(θh)P k F − E〈DG, h〉 [0,T ] P k Fk=0k=0= EG(θh)P n−1 F + EG(θh)P n+1 F − E〈DG, h〉 [0,T ] P n−1 F.By the integration-by-parts formula of Corollary 4.3.11,EG(θh)P n−1 F − E〈DG, h〉 [0,T ] P n−1 F = EG〈DP n−1 F, h〉 [0,T ] ,which is zero since G ∈ H ′′ n and 〈DP n−1 F, h〉 [0,T ] ∈ H n−2 by Lemma 4.4.19. Since alsoE〈DG, h〉 [0,T ] P n+1 F = 0 by the same lemma, we can use Corollary 4.3.11 once againto obtainEG(θh)P n−1 F + EG(θh)P n+1 F − E〈DG, h〉 [0,T ] P n−1 F= EG(θh)P n+1 F= EG(θh)P n+1 F − E〈DG, h〉 [0,T ] P n+1 F= EG〈DP n+1 F, h〉 [0,T ]= 〈G ⊗ h, DP n+1 F 〉.All in all, we have proven for G ∈ H ′′ n that〈G ⊗ h, DF 〉 = 〈G ⊗ h, DP n+1 F 〉.Step 2: Conclusions. Noting that both the left-hand side and the right-hand sidein the equality above is continuous and linear in G, we can extend it to hold notonly for G ∈ H n, ′′ but also for G ∈ H n . Since the variables of the form G ⊗ hfor G ∈ H n and h ∈ L 2 [0, T ] has dense span in H n (Π) by definition, we conclude,again by continuity and linearity of the inner product, that 〈Z, DF 〉 = 〈Z, DP n+1 F 〉for all Z ∈ H n (Π). Since DP n+1 F ∈ H n (Π) by Corollary 4.4.17, Lemma A.6.18finally yields that DP n+1 F is the orthogonal projection of DF onto H n (Π), that is,Pn Π DF = DP n+1 F .Theorem 4.4.22. Let F ∈ L 2 (F T ). F is in D 1,2 if and only if ∑ ∞n=1 n‖P nF ‖ 2 isfinite. In the affirmative case, DF = ∑ ∞n=0 DP nF , and in particular the norm is givenby ‖DF ‖ 2 = ∑ ∞n=1 n‖P nF ‖ 2 .Proof. First assume that F ∈ L 2 (F T ) and that ∑ ∞n=1 n‖P nF ‖ 2 2 < ∞. We have toshow that F ∈ D 1,2 and identify DF and ‖DF ‖ 2 . By Lemma A.6.12, F = ∑ ∞n=0 P nF .

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 171Define F k = ∑ kn=0 P nF , then F k converges to F in L 2 (F T ). We will argue that DF kconverges as well, this will imply F ∈ D 1,2 .Obviously, P n F ∈ H n , so by Corollary 4.4.16, P n F ∈ D 1,2 and ‖DP n F ‖ 2 = n‖P n F ‖ 2 .Therefore, we find ∑ ∞n=0 ‖DP nF ‖ 2 = ∑ ∞n=0 n‖P nF ‖ 2 < ∞, showing that the series∑ ∞n=0 DP nF is convergent in L 2 (F T ), meaning that the sequence DF n is convergent.By closedness of D, we conclude F ∈ D 1,2 and DF = ∑ ∞n=0 DP nF . By Corollary4.4.18, DP n F and DP m F are orthogonal whenever n ≠ m. We may therefore alsoconclude ‖DF ‖ 2 = ∑ ∞n=0 ‖DP nF ‖ 2 = ∑ ∞n=1 n‖P nF ‖ 2 .It remains to show that ∑ ∞n=1 n‖P nF ‖ 2 < ∞ is also necessary to obtain F ∈ D 1,2 .Assume that F ∈ D 1,2 , we need to prove that the sum is convergent. To this end, notethat using the orthogonal decomposition of L 2 (Π) from Theorem 4.4.13 and applyingLemma A.6.12 and Theorem 4.4.21,∞∑∞∑∞∑DF = Pn Π DF = DP n+1 F = DP n F.n=0n=0In particular, the sum on the right is convergent, and therefore ∑ ∞n=1 ‖DP nF ‖ 2 is finite.By Corollary 4.4.16, ‖DP n F ‖ 2 = n‖P n F ‖ 2 , and we conclude that ∑ ∞n=1 n‖P nF ‖ 2is finite, as desired.n=1Theorem 4.4.22 is the promised characterisation of D 1,2 in terms of the subspaces H n .As our final result on the Malliavin calculus, we will show how the theorem can beused to obtain an extension of the chain rule for Lipschitz mappings. We are going toemploy some weak convergence results, see Appendix A.6 for an overview.Lemma 4.4.23. Let F n be a sequence in D 1,2 converging to F and assume that DF nis bounded in L 2 (Π). Then F ∈ D 1,2 , and there is a subsequence of DF n convergingweakly to DF .Proof. We will use Theorem 4.4.22 to argue that F ∈ D 1,2 . To do so, we need to provethat ∑ ∞m=1 m‖P mF ‖ 2 is finite. To this end, first note that since DF n is boundedin L 2 (Π), by Theorem A.6.21 there exists a subsequence DF nk converging weakly tosome α ∈ L 2 (Π). Convergence of F nk to F and Corollary 4.4.16 allows us to conclude∞∑m‖P m F ‖ 2 =m=1∞∑lim m‖P m F nk ‖ 2 =km=1∞∑lim ‖DP m F nk ‖ 2 .km=1

172 The Malliavin CalculusNow note that P m F nk is a sequence in H m which converges to P m F since P m iscontinous, being an orthogonal projection. By Corollary 4.4.16, D is a continuousmapping on H m , so we find that DP m F nk converges to DP m F . On the other hand,by Theorem 4.4.21, DP m F nk = Pm−1DF Π nk . Since DF nk converges weakly to α,Pm−1DF Π nk converges weakly to Pm−1α Π by Lemma A.6.23. Since ordinary convergenceimplies weak convergence by Lemma A.6.20 and weak limits are unique by LemmaA.6.19, we obtain DP m F = Pm−1α. Π All in all, we conclude that DP m F nk convergesto Pm−1α. Π Therefore, the norms converge as well. Thus, using Lemma A.6.12,∞∑lim ‖DP m F nk ‖ 2 =km=1∞∑‖Pm−1α‖ Π 2 = ‖α‖ 2 ,which is of course finite. Theorem 4.4.22 now yields F ∈ D 1,2 . Finally, note thatP Π mDF = DP m+1 F = P Π mα for any m ≥ 0, so by Lemma A.6.12, DF = α. Thismeans that DF nk converges weakly to DF .Corollary 4.4.24. Let F n be a sequence in D 1,2 converging to F and assume thatDF n is bounded in L 2 (Π). Then F ∈ D 1,2 , and DF n converges weakly to DF .m=1Proof. By Lemma 4.4.23, F ∈ D 1,2 . We need to show that DF n converges weakly toDF . To this end, it will suffice to show that for any subsequence DF nk , there is afurther subsequence DF nkl converging to DF . Therefore, let a subsequence DF nk begiven. Then F nk converges to F and DF nk is bounded in L 2 (Π), so Lemma 4.4.23 showsthat there is a subsequence F nkl converging to DF . This concludes the proof.Theorem 4.4.25 (Lipschitz chain rule). Let F ∈ D n 1,2, F = (F 1 , . . . , F n ), and letϕ : R n → R be a Lipschitz mapping with Lipschitz constant K with respect to ‖ · ‖ ∞ .Then ϕ(F ) ∈ D 1,2 , and there is a random variable G = (G 1 , . . . , G n ) with values inR n and ‖G k ‖ 2 ≤ K such that Dϕ(F ) = ∑ nk=1 G kDF k .Proof. By Lemma A.4.2, there exists a sequence of mappings ϕ n ∈ C ∞ (R n ) withpartial derivatives bounded by K converging uniformly to ϕ. By the ordinary chain ruleof Theorem 4.3.1, Dϕ n (F ) = ∑ n ∂ϕ nk=1 ∂x k(F )DF k . Now, since ϕ n converges uniformlyto ϕ, ϕ n (F ) converges in L 2 (F T ) to ϕ(F ). On the other hand, we haven∑∥so the sequence ∑ n ∂ϕ nk=1ϕ(F ) ∈ D 1,2 and ∑ nk=1k=1∂ϕ n∂x k(F )DF k∥ ∥∥∥∥2≤ Kn∑‖DF k ‖ 2 ,k=1∂x k(F )DF k is bounded in L 2 (Π). By Corollary 4.4.24 we obtain∂ϕ n∂x k(F )DF k converges weakly to Dϕ(F ). If we can identify

4.4 Orthogonal subspaces of L 2 (F T ) and L 2 (Π) 173the weak limit as being of the form stated in the theorem, we are done. To this end,note that since the sequence ∂ϕ n∂x k(F ) is pointwisely bounded by K, it is bounded inL 2 (F T ) by K as well. Therefore, by Theorem A.6.21 there exists a subsequence suchthat for all k ≤ n, ∂ϕ nm∂x k(F ) converges weakly to some G k ∈ L 2 (F T ). From LemmaA.6.22 we know that ‖G k ‖ 2 ≤ K, but we would like a pointwise bound instead. Toobtain this, let A ∈ F T . We then obtainE1 A G 2 k = 〈1 A G k , G k 〉〈= lim 1 A G k , ∂ϕ 〉n m(F )m ∂x k= E1 A G k∂ϕ nm∂x k(F )≤ K1 A |G k |.This shows in particular that G 2 k≤ K|G k| almost surely, and therefore |G k | ≤ Kalmost surely. Now, for any bounded element X of L 2 (Π), we find〈X, ∂ϕ n m∂x k(F )DF k − G k DF k〉〈 ( ) 〉∂ϕnm= X, (F ) − G k DF k∂x k( ∫ T( ) )∂ϕnm= E X(t) − G k DF k (t) dt0 ∂x k( ∫ )T(∂ϕnm )= E X(t)DF k (t) dt − G k .∂x k0Since X is bounded and DF k ∈ L 2 (Π), we conclude by Jensen’s inequality that∫ T0 X(t)DF k(t) dt ∈ L 2 (F T ). By the weak convergence of ∂ϕ nm∂x kto G k , the abovetherefore tends to zero. We may now conclude, still letting X ∈ L 2 (Π) be bounded,〈X, Dϕ(F ) −n∑〉n∑G k DF k = lim〈X,mk=1k=1∂ϕ nm∂x k(F )DF k −n∑k=1∂ϕ nm∂x k(F )DF k〉= 0.Since the bounded elements of L 2 (Π) are dense, by Lemma A.6.2 and A.6.13, weconclude Dϕ(F ) = ∑ nk=1 G kDF k . This proves the theorem.This concludes our exposition of this Malliavin calculus. In the final next section, wegive a survey of the further theory of the Malliavin calculus.

174 The Malliavin Calculus4.5 Further theoryOur description of the Malliavin calculus in the preceeding sections only covers thebasics of theory, and is in fact quite inadequate for applications. We will now describesome of the further results of the theory and outline the main theoretical areas whereMalliavin calculus finds application. All of these areas are covered in more or lessdetail in Nualart (2006).The Skorohod integral. The largest omission in our exposition is clearly that ofthe Skorohod integral. The Skorohod integral δ is defined as the adjoint operator ofthe Malliavin derivative D. This is a linear operator defined on a dense subspace S 1,2of L 2 (Π) mapping into L 2 (F T ), characterised by the duality relationship〈F, δu〉 FT= 〈DF, u〉 Πfor any u ∈ S 1,2 . The existence of such an operator follows from the closedness ofD combined with Lemma 19.2 and Proposition 19.5 of Meise & Vogt (1997). TheSkorohod integral has two extremely important properties which connect it to thetheory of stochastic integration. First off, if u ∈ L 2 (Π) is progressively measurable, itholds thatδ(u) =∫ T0u s dW s ,where the integral on the right is the ordinary Itô stochastic integral. This justifies thename “Skorohod integral” for δ, and shows that the Skorohod integral is an extensionof the Itô stochastic integral. Inspired by this fact, we will use the notation ∫ T0 u sδW sfor δ(u). The second important property is as follows. Writing D t F = (DF ) t , if uis such that u t ∈ D 1,2 for all t ≤ T and such that there is a measurable version of(t, s) ↦→ D t u s and a measurable version of ∫ T0 D tu s δW s , then∫ TD t u s δW s =0∫ T0D t u s δW s + u t .In the case where u is progressively measurable, this shows how to differentiate ageneral Itô stochastic integral.The properties of the Skorohod integral are usually investigated through the Itô-Wienerexpansion, which is a series expansion for variables F ∈ L 2 (F T ). The basic contentof the expansion is, with ∆ n = {t ∈ [0, T ] n |t 1 ≤ · · · ≤ t n }, that there exists squareintegrabledeterministic functions f n on ∆ n such that∞∑∫ T ∫ tn∫ t3∫ t2F = n! · · · f n (t) dW (t 1 ) · · · dW (t n ),n=00000

4.6 Notes 175where the convergence is in L 2 . Expanding a process on [0, T ] in this manner, one maythen identify a necessary and sufficient criterion on the mappings f n to ensure that theprocess is adapted. This can be used to prove the relationship between the Skorohodintegral and the Itô integral. All of this is described in Section 1.3 of Nualart (2006).Differentiation of SDEs. Being a stochastic calculus, it seems obvious that theMalliavin calculus should yield useful results when applied to the theory of stochasticdifferential equations. The main result on this topic is described in Section 2.2 ofNualart (2006) and basically states that under suitable regularity conditions, whentaking the Malliavin derivative of the solution to a stochastic differential equation,interchange of differentiation and integration is allowed. This result is of fundamentalimportance to the applications in financial mathematics.Regularity of densities. The Malliavin calculus can also be used to give abstractcriteria for when a random variable possesses a density with respect to the Lebesguemeasure, and criteria for when such a density possesses some degree of smoothness.This is shown in Section 2.1 of Nualart (2006). The basic results are Theorem 2.1.1and Theorem 2.1.4, yielding sufficient criteria in terms of the Malliavin derivativesof a vector variable for the existence and smoothness of a density with respect tothe Lebesgue measure. A further important result is found as Theorem 2.3.3, givinga sufficient criterion for the solution of a stochastic differential equation to have asmooth density.4.6 NotesWe will in this section review our results and compare them to other accounts of thesame results in the theory.Our exposition of the theory is rather limited in its reach, striving primarily for rigorand detail. The theory is based almost entirely on Section 1.1 and Section 2.1 ofNualart (2006). Nualart (2006) does not always provide many details, so the virtueof our results is mostly the added detail. The main difficulties has been to providedetails of the chain rule of Theorem 4.3.1 and the relations between the orthogonaldecompositions of L 2 (F T ) and L 2 (Π), culminating in the proof of Theorem 4.4.21.Further work has gone into the extension of the chain rule given in Theorem 4.3.5,which hopefully will be useful in the later development of the theory.

176 The Malliavin CalculusThe Malliavin calculus is a relatively young theory, originating in the 1970s, and eventhough the recent applications in finance such as the results in Fournié et al. (1999)seem to have motivated an increase in production of material related to the theory,there are still quite few sources for the theory.The most well-known book on the Malliavin calculus is Nualart (2006), which as notedhas been the fundamental source for all of the theory described in this chapter. Otherbooks on the topic are Bell (1987), Bass (1998), Malliavin (1997) and Sanz-Solé(2005).In general, these books are all, to put it bluntly, either not very rigorous or somewhatinaccessible. Nualart (2006) is clearly the best exposition of these, but is still difficultcompared to the literature available for, say, stochastic integration. We will nowdiscuss the merits and demerits of each book.The book by Bell (1987) is very small, and in general either defers proofs to otherworks or only gives sketches of proofs, making it difficult to use as an introduction.The book Bass (1998) does not have the Malliavin calculus as its main topic, andonly spends the final chapter exploring it. Considering this, it cannot be blamed fornot developing the theory in detail. Still, it is not useful as an introductory work.The work Sanz-Solé (2005) looks very useful at a first glance, with a nice layout anda seemingly pedagogical introduction. However, it also defers many proofs to otherbooks and often entirely skips proofs. This work cannot be recommended as an goodintroduction either.Another work on the Malliavin calculus is Malliavin (1997). This book is not onlyabout the Malliavin calculus, but touches upon a good deal of different topics. In anycase, the main problem with the book is that it is so impressively abstract that itseems almost made to be incomprehensible on purpose.This leaves Nualart (2006). While it, as noted earlier, is not very accessible, it isclearly superior to the other books. It is cast in a modern manner, and does not makethe theory unnecessarily abstract. Our account differs from that one by considering aone-dimensional Brownian motion instead of what is known as an isonormal gaussianprocess. The approach based on isonormal processes restricts the theory to the contextof Hilbert spaces. This would make the introductory results of Section 4.2 prettier,but would also make motivation for the definitions more difficult to comprehend.

4.6 Notes 177In general, what makes the book difficult is that it very often omits a great deal ofdetail. This scope of these omissions can be estimated by noting that the theory whichhave been developed in 40 pages here and at least 15 pages more in the appendix isdeveloped in only 10 pages in Nualart (2006).Complementing these books, there are also notes available on the internet documentingthe Malliavin calculus. Some of these are Zhang (2004), Bally (2003), Friz (2002)and Øksendal (1997). Of these, only Øksendal (1997) is useful.The notes Bally (2003) and Friz (2002) are very easy to identify as useless. Bothengage in what best can be described as some sort of mathematical name-dropping,often invoking Sobolev spaces, distribution theory and other theories, without theseinvocations ever resulting in any actual proofs. Furthermore, proofs are in generalheuristic or omitted. Zhang (2004) at a first glance looks very good. As with Sanz-Solé (2005), the layout is professional and the author quickly obtains an air of havinga good overview of the theory. However, on closer inspection most of the proofs haveconspicuously similar levels of detail as in Nualart (2006), thereby not really addinganything new.The final note we discuss is Øksendal (1997). This is by far the most accessible textavailable on the Malliavin calculus. It develops the theory in manner very differentfrom Nualart (2006), basing the definition of the Malliavin derivative and Skorohodintegral directly on the Wiener-Itô expansion. The proofs are in general given in detailand are very readable. Unfortunately, the note is short and does not cover so muchof the theory, so at some point one has to revert to other works. However, as anintroduction, it is most definitely very useful.Complementing these works is the forthcoming book Di Nunno et al. (2008). Itextends the theory of the Malliavin calculus to processes other than Brownian motion,but does also cover the Brownian case in detail. Even though the final work hereis based on Nualart (2006), Di Nunno et al. (2008) has been invaluable as a firstintroduction, both to the Malliavin calculus in general and also for the applications tofinance, because of its high level of detail and readability.

178 The Malliavin Calculus

Chapter 5Mathematical FinanceIn this chapter, we will develop the theory of arbitrage pricing for a very basic classof models for financial markets. Our goals is to find sufficient criteria for the marketsto be free of arbitrage, to apply these criteria to some simple models and to show howto calculate prices and sensitivities in these models.For ease of notation, we will in general use the shorthand dX t = µ dt + σ dW t forX t = X 0 + ∫ t0 µ t dt + ∫ t0 σ t dW t for some X 0 .5.1 Financial market modelsWe work in the context of a filtered probability space (Ω, F, P, F t ) with a n-dimensionalF t Brownian motion W . We assume that the usual conditions hold, in particular weassume that F t is the usual augmentation of the filtration generated by W . Furthermore,we assume that F = F ∞ = σ(∪ t≥0 F t ).We begin by defining the fundamental building blocks of what will follow. We willassume given a one-dimensional stochastic short rate process r and define the riskfreeasset price process B as the solution to dB t = r t B t dt with B 0 some positiveconstant, equivalent to putting B t = B 0 exp( ∫ t0 r s ds). We furthermore assume givenm financial asset price processes (S 1 , . . . , S m ), which we assume to be nonnegative

180 Mathematical Financestandard processes.Together, the short rate, the risk-free asset price process and the financial asset priceprocesses define a financial market model M. In this market, a portfolio strategy is apair of locally bounded progressive processes h = (h 0 , h S ), where h 0 is one-dimensionaland h S is m-dimensional. The interpretation is that h 0 (t) denotes the number of unitsheld at time t in the risk-free asset B, and h S k(t) denotes the number of units heldat time t in the k’th risky asset S k . To a given portfolio strategy, we associate thevalue process V h , defined by V h (t) = h 0 (t)B(t) + ∑ mk=1 hS k (t)S k(t). We say that h isadmissible if V h is nonnegative. We say that the portfolio h is self-financing if it holdsthatm∑dV h (t) = h 0 (t) dB(t) + h S k (t) dS k (t).Here, the integrals are always well-defined because h is assumed to be locally boundedand progressive. The above essentially means that the change in value in the portfolioonly comes from profits and losses in the portfolio, there is no exogenous intake orouttake of wealth. That this is the correct criterion for no exogenous infusions ofwealth is not as obvious as it may seem. It is important that we are using the Itôintegral and not, say, the Stratonovich integral. For an argument showing that the Itôintegral yields the correct interpretation, see Chapter 6 of Björk (2004) and comparewith the Riemann approximations for the Itô and Stratonovich integrals of LemmaIV.47.1 and Lemma IV.47.3 in Rogers & Williams (2000b).k=1We are now ready to make the central definition of this section, the notion of anarbitrage opportunity.Definition 5.1.1. We say that an admissible and self-financing portfolio is an arbitrageopportunity at time T if the value process V h satisfies V0 h = 0, P (VT h ≥ 0) = 1and P (VTh > 0) > 0. If a market M contains no arbitrage opportunities at time T , wesay that it is T -arbitrage free. If a market contains no arbitrage opportunities at anytime, we say that it is arbitrage free.Comment 5.1.2 The notion of being arbitrage free is the basic requirement for amarket to be realistic. It it reasonable to state that most real-world markets arearbitrage-free most of the time. Therefore, if we are considering a market which hasarbitrage, we are some distance away from anything useful for real-world modeling.The three requirements for h to constitute an arbitrage opportunity can be put inwords as:

5.1 Financial market models 1811. It must be possible to enter the strategy without cost.2. The strategy should never result in a loss of money.3. The stragegy should have a positive probability of resulting in a profit.Note that for a portfolio to be an arbitrage opportunity, it must be both self-financingand admissible. Clearly, a portfolio satisfying the three requirements above but whichis not self-financing does not say anything about whether our market is realistic, anyonecan generate a sure profit if exogenous money infusions are allowed. The reason wealso require admissibility is to rule out possibilities for “doubling schemes” - borrowingmoney until a profit turns out. For more on doubling schemes, see Steele (2000),Section 14.5 or Example 1.2.3 of Karatzas & Shreve (1998).◦We are interested in finding a sufficient criterion for a market to be arbitrage free.To formulate our main result on this, we first need to introduce the concept of anormalized market. Given our financial instrument vector S and the risk-free asset B,we define the normalized instruments by Sk ′ (t) = S k(t)B(t) . The instruments S′ kcan bethought of as the discounted versions of S k .Lemma 5.1.3. S ′ kis a standard process with the dynamicsdS ′ k(t) = 1B(t) dS k(t) − S ′ k(t)r(t) dt.Proof. Since B is positive, we can use Itô’s lemma in the form of Corollary 3.6.4 onthe C 2 (R 2 \ {0}) mapping (x, y) ↦→ x yand obtaindS ′ k(t) ===1B(t) dS k(t) − S k(t)B(t) 2 dB(t) − 12B(t) 2 d[S k, B] t + S k(t)B(t) 3 d[B] t1B(t) dS k(t) − S k(t)r(t)B(t) dtB(t)21B(t) dS k(t) − S k(t)r(t) ′ dt,as desired.Lemma 5.1.3 shows that the processes S ′ = (S ′ 1, . . . , S ′ m) also describe financial instrumentsin the sense introduced earlier in this section, the processes are nonnegativestandard processes. In particular, we can consider the normalized market consisting

182 Mathematical Financeof a trivial risk-free asset B ′ = 1 and the normalized instruments S ′ . Our next lemmashows that to check arbitrage for the general market, it will suffice to consider thenormalized market.Lemma 5.1.4. The market with instruments S and risk-free asset B is T -arbitragefree if the normalized market with instruments S ′ and risk-free asset B ′ = 1 is T -arbitrage free.Proof. Assume that the normalized market is free of arbitrage at time T . Let theportfolio h = (h 0 , h S ) be an arbitrage opportunity at time T for the original market.We will work to obtain a contradiction. Let VSh be the value process when consideringh as a portfolio strategy for the original market. Let VS h be the value process when′considering h as a portfolio strategy for the normalized market. Our goal is to showthat the portfolio h also is an arbitrage under the normalized market.We first examine VS h ′. Note that since B′ = 1, we obtain()m∑VS h ′(t) = h0 (t) + h S k (t)S k(t) ′ = 1m∑h 0 (t)B(t) + h S k (t)S k (t)B(t)k=1k=1= V h S (t)B(t) .In particular, since B is positive, we find that VS h is nonnegative, so h is admissible′under the normalized market. To show that it is self-financing, we use Itô’s lemmaand Lemma 5.1.3 to obtaindVS h ′(t) = 1B(t) dV S h (t) − V S h(t)B(t) 2 dB(t) − 12B(t) 2 d[V S h , B] t + V S h(t)B(t) 3 d[B] tm∑====1B(t) h0 (t) dB(t) + 1B(t)1B(t)1B(t)m∑k=1k=1k=1h S k (t) dS k (t) − V h S ′(t)B(t) dB(t)m∑h S k (t) dS k (t) − 1 (VhB(t) S ′(t) − h 0 (t) ) dB(t)m∑m∑h S k (t) dS k (t) − h S k (t)S k(t)r(t) ′ dtk=1h S k (t) dS ′ k(t),so h is self-financing under the normalized market. It remains to show that h isan arbitrage opportunity under the normalized market. Since h is an arbitrage forthe original market, VS h(0) = 0, P (V S h(T ) ≥ 0) = 1 and P (V S h (T ) > 0) > 0. Nownote that since B(t) > 0 for all t ≥ 0, we obtain VS h ′(0) = 0, P (V S h ′(T ) ≥ 0) = 1k=1

5.1 Financial market models 183and P (VS h ′(T ) > 0) > 0. We conclude that h is an arbitrage opportunity under thenormalized market. Since this market was assumed arbitrage free, we have obtained acontradiction and must conclude that the original market is arbitrage free.Next, we prove our main results for a market to be free of arbitrage. The theorembelow is the primary criterion, with its corollaries giving criteria for special cases whichare easier to check in practice. We say that a proces M is a local martingale on [0, T ]if there exists a localising sequence τ n such that M τ nis a martingale on [0, T ] for eachn.Theorem 5.1.5. Let T > 0. If there exists a probability measure Q equivalent toP such that each instrument of the normalized market is a local martingale on [0, T ]under Q, then M is arbitrage-free at time T .Proof. By Lemma 5.1.4, it will suffice to show that the normalized market is arbitragefree at time T . Therefore, assume that h is an arbitrage opportunity at time T underthe normalized market with value process VS h ′, we will aim to obtain a contradiction.Let Q be the probability measure equivalent to P which makes Sk ′ a local martingaleon [0, T ] for any k ≤ m. By definition of an arbitrage opportunity, we knowVS h ′(0) = 0,P (VS h ′(T ) ≥ 0) = 1,P (VS h ′(T ) > 0) > 0.Since P and Q are equivalent, we also have Q(VS h ′(T ) ≥ 0) = 1 and Q(V S h ′(T ) > 0) > 0.In particular, E Q VS h ′(0) = 0 and EQ VS h ′(T ) > 0. We will argue that h cannot be anarbitrage opportunity by showing that VS h ′ is a supermartingale on [0, T ] under Q,contradicting that E Q VS h ′(0) < EQ VS h ′(T ).Under the normalized market, the price of the risk-free asset is constant, so we finddVS h ′(t) = ∑ mk=1 hS k (t) dS′ k (t). Now, since P and Q are equivalent, S′ kis a standardprocess under Q according to Theorem 3.8.4. Because h is locally bounded, h isintegrable with respect to Sk ′ both under P and Q, and Lemma 3.8.4 then yields thatthe integrals are the same whether under P or Q.Under Q, each Sk ′ is a continuous local martingale on [0, T ]. Therefore, the Q stochasticintegral process ∑ n∫ tk=1 0 hS k (s) dS′ k(s) is a continuous local martingale on [0, T ] underQ. Under P , the stochastic integral process ∑ n∫ tk=1 0 hS k (s) dS′ k (s) is equal to V S h ′(t).

184 Mathematical FinanceSince the integrals agree, as noted above, we conclude that VS h is a continuous local′martingale on [0, T ] under Q. Because h is admissible, VS h ′ is nonnegative. Thus,under Q, VS h is a nonnegative continuous local martingale on [0, T ]. Using the same′argument as in the proof of Lemma 3.7.2, V h is then a Q supermartingale on [0, T ]. Inparticular, E Q VS h ′(t) is decreasing on [0, T ], in contradiction with our earlier conclusionthat E Q VS h ′(0) < EQ VS h ′(T ). Therefore, there can exist no arbitrage opportunities inthe normalized market. The result follows.Comment 5.1.6 Note that it was our assumption that h is locally bounded whichensured that the integral of h both under P and Q was well-defined. This is thereason why we in our definitions have chosen only to consider locally bounded portfoliostrategies. The equivalent probability measure in the theorem making each instrumentof the normalized market a local martingale on [0, T ] is called an equivalent localmartingale measure for time T , or a T -EMM.◦The next two corollaries provide sufficient criteria for being arbitrage-free in the specialecase of a linear financial market, which we now define.Definition 5.1.7. A financial market is said to be linear if each S k has dynamicsdS k (t) = µ k (t)S k (t) dt +n∑σ ki (t)S k (t) dW i (t)where σ ki ∈ L 2 (W ), µ k ∈ L 1 (t), and S k (0) is some positive constant. Here, µ is knownas the drift vector process and σ is the volatility matrix process.i=1By Lemma 3.7.1, the price processes in a linear market has the explicit form( ∫ tn∑∫ tS k (t) = S k (0) exp µ k (s) ds + σ ki (s) dWs i − 1 n∑∫ )tσ 22ki(s) ds ,0i=1unique up to indistinguishability. In particular, S is nonnegative, so it actually definesa financial market in the sense introduced earlier.Corollary 5.1.8. Assume that the market is linear and that there exists a process λ inL 2 (W ) n such that µ k (t) − r(t) = ∑ ni=1 σ ki(t)λ i (t) for any k ≤ m, λ ⊗ P almost surely.Assume further that with M t = − ∑ ni=1∫ t0 λ i(s) dW i s, E(M) is a martingale. Thenthe market is free of arbitrage, and there is a T -EMM Q such that the Q-dynamics on[0, T ] are given by dS k (t) = r(t)S k (t) dt + ∑ ni=1 σ ki(t)S k (t) dW i (t), where W is a F tBrownian motion under Q.0i=10

5.1 Financial market models 185Proof. We will use Theorem 5.1.5. Let T > 0 be given. Since E(M) is a martingaleand E(M T ) = E(M) T , we obtain EE(M T ) ∞ = EE(M) T = 1. Recalling that F = F ∞ ,we can then define the measure Q on F by Q(A) = E P 1 A E(M T ) ∞ . Since E(M T ) ∞is almost surely positive and has unit mean, P and Q are equivalent. Now, with= −λ i (t)1 [0,T ] (t), we obtain M T = Y · W . Therefore, by the Girsanov Theorem ofTheorem 3.8.3, we can define W by W k t = Wtk + ∫ t0 λ k(s)1 [0,T ] (s) ds and obtain thatW is an F t Brownian motion under Q.Y itWe claim that under Q, Sk ′using Lemma 5.1.3,is a local martingale on [0, T ]. To this end, we note that,===S ′ k(t) − S ′ k(0)∫ t0∫ t0∫ t(µ k (s) − r(s))S ′ k(s) ds +((µ k (s) − r(s) −n∑∫ ti=1 0σ ki (s)S ′ k(s) dW i s)n∑σ ki (s)λ i (s)1 [0,T ] (s) S k(s) ′ ds +i=1(µ k (s) − r(s))1 (T,∞) (s)S ′ k(s) ds +n∑∫ t0i=1 0n∑∫ ti=1 0σ ki (s)S ′ k(s) dW i s,σ ki (s)S ′ k(s) dW i swhere we have used that ∑ ni=1 σ ki(s)λ i (s)1 [0,T ] (s) = (µ k (s) − r(s))1 [0,T ] (s). Lettingτ n be a determining sequence making the latter integral above into a martingale, it isthen clear that (Sk ′ )τn is a martingale on [0, T ] under Q. Thus, Sk ′ is a local martingaleon [0, T ] under Q, so Q is a T -EMM. By Theorem 5.1.5, the market is free of arbitrage.Using the proof technique of Lemma 5.1.3, we find that the Q-dynamics of S k on [0, T ]areas desired.dS k (t) = B(t) dS k(t) ′ + S k (t)r(t) dt=n∑σ ki (t)S k (t) dW i k + S k (t)r(t) dt,i=1Comment 5.1.9 In the above, our assumption that F = F ∞ became important. Hadwe not known that F = F ∞ , we would only have obtained a measure on F ∞ equivalentto the restriction of P to F ∞ , and not a measure on all of F.The equation µ k (t)−r(t) = ∑ ni=1 σ ki(t)λ i (t) is called the market price of risk equations,and any solution λ is known as a market price of risk specification for the market. In

186 Mathematical Financethe one-dimensional case, we have a unique solution λ t = µ t−r tσ t, showing that λ canbe interpreted as the excess return per volatility. Therefore, λ can in an intuitive sensebe said to measure the price of uncertainty, or risk, in the market.◦Corollary 5.1.10. Assume that the market is linear and only has one asset. Letµ denote the drift and let σ denote the volatility. Assume that the drift and shortrate are constant, and that σ is positive λ ⊗ P almost surely. If there is k ≤ n suchthat E exp( (µ−r)22∫ t01σ k (s) 2 ds) is finite for any t ≥ 0, the market is free of arbitrage,and there is a T -EMM such that the Q-dynamics on [0, T ] are given on the formdS k (t) = r(t)S k (t) dt + ∑ ni=1 σ ki(t)S k (t) dW i (t), where W is a F t Brownian motionunder Q.Proof. We want to solve the market price of risk equations of Corollary 5.1.8. Sincewe only have one asset, they reduce ton∑µ − r = σ i (t)λ i (t).Let k be such that E exp( (µ−r)22∫ t0i=11σ k (s) 2 ds) is finite for any t ≥ 0. To simplify, wedefine λ i = 0 for i ≠ k. We are then left with the equation µ − r = σ k (t)λ k (t).Let A denote the subset of [0, ∞) × Ω where σ is positive. Since σ is progressive,A ∈ Σ π . Defining λ k (t, ω) = µ−rσ k (t,ω)whenever (t, ω) ∈ A and zero otherwise, λsatisfies the equations in the statement of Corollary 5.1.8. By that same corollary,in order for the market to be arbitrage free, it will then suffice to show that withM t = − ∑ ni=1∫ t0 λ i(s) dW i s, E(M) is a martingale, and to do so it will suffice to showthat E(M t ) is a uniformly integrable martingale for any t ≥ 0. By the Novikov criterionof Theorem 3.8.10, this is the case if E exp( 1 2 [M t ] ∞ ) is finite for any t ≥ 0. But wehave( ) 1E exp2 [M t ] ∞= E exp( 12 [M] t( 1= E exp2∫ t0)( (µ − r)2= E exp2)λ k (s) 2 ds∫ t0)1σ k (s) 2 ds ,so that the Novikov criterion is satisfied follows directly from our assumptions. Theexistence of a Q measure with the given dynamics then follows from Corollary 5.1.8.We are now done with exploring sufficient criteria for no-arbitrage. Next, we considerpricing of claims in markets without arbitrage. For T > 0, we define a T -claim as a

5.1 Financial market models 187F T measurable, integrable variable X. The intuition behind this is that the owner ofthe claim receives the amount X at time T . Our goal is to argue for a reasonable priceof this contract at t ≤ T . We will not be able to do this in full generality, but willonly consider markets satisfying the criterion of Theorem 5.1.5. As discussed in thenotes, this actually covers all arbitrage-free markets, but we will not be able to showthis fact.Theorem 5.1.11. Let T > 0 and consider a market with a T -EMM Q. Let X be anyT -claim. Assume that X ≥ 0 and define( ∣ )B(t) ∣∣∣S m+1 (t) = E Q B(T ) X F t 1 [0,T ] (t) + B(t)B(T ) X1 (T,∞)(t).The market with instruments (S 1 , . . . , S m+1 ) is free of arbitrage at time T .Proof. We first argue that (S 1 , . . . , S m+1 ) is a well-defined financial market. Under Q,S ′ m+1 is obviously a martingale. Since we are working under the augmented Brownianfiltration, it has a continuous version. Since X is nonnegative, S ′ m+1(t) is almost surelynonnegative for ant t ≥ 0, and therefore there is a version of S ′ m+1 which is nonnegativein addition to being continuous. Thus, S ′ m+1 is a nonnegative standard process underQ. Therefore, S m+1 is also a nonnegative standard process under Q, and so it is astandard process under P . Finally, we conclude that (S 1 , . . . , S m+1 ) defines a financialmarket.Under Q, each of the instruments S ′ 1, . . . , S ′ m are local martingales on [0, T ]. SinceS ′ m+1 trivially is a martingale on [0, T ] under Q, Theorem 5.1.5 yields that the marketwith instruments (S 1 , . . . , S m+1 ) is free of arbitrage at time T .Comment 5.1.12 The somewhat cumbersome form of S m+1 , splitting up into a parton [0, T ] and a part on (T, ∞), is an artefact of considering markets with infinitehorizon.◦What Theorem 5.1.11 shows is that for any nonnegative claim, using E Q ( B(t)B(T ) X|F t)as the price for the claim X for t ≤ T is consistent with no arbitrage in the marketwhere the claim is traded. And since E Q ( B(T )B(T ) X|F T ) = X, buying the asset with priceprocess E Q ( B(t)B(T ) X|F t) actually corresponds to buying the claim, in the sense that atthe time of expiry T , one owns an asset with value X. Assuming that prices are linear,we conclude that a reasonable price for any T -claim for t ≤ T is E Q ( B(t)B(T ) X|F t), whereQ is a T -EMM.

188 Mathematical FinanceWe are now done with our discussion of general financial markets. Our results are thefollowing. We have in Theorem 5.1.5 obtained general criteria for a market to be freeof arbitrage, and we have obtained specialised criteria for the case of linear markets inthe corollaries. Afterwards, we have shortly argued, based on the result of Theorem5.1.11, what a reasonable price of a contingent claim should be modeled as. Note thatlack of uniqueness of T -EMMs can imply that the price of a T -claim is not uniquelydetermined. This will not bother us particularly, however.We will now proceed to discuss the practical matter of calculating the prices andsensitivities associated with contingent claims. We will begin by a very short reviewof some basic notions from the theory of SDEs and afterwards proceeed to go throughsome methods for Monte Carlo evaulation of prices and sensitivities. Having done so,we will apply these techniques to the Black-Scholes and Heston models.5.2 Stochastic Differential EquationsWe will review some fundamental definitions regarding SDEs of the formdX k t = µ k (X t ) dt +n∑σ ki (X t ) dWt i ,where X 0 is constant, k ≤ m, W is a n-dimensional Brownian motion and the mappingsµ : R m → R m and σ : R m → R m×n are measurable. We also consider a deterministicinitial condition x ∈ R m . We write the SDE in the shorthandi=1dX t = µ(X t ) dt + σ(X t ) dW t ,the underlying idea being that the m-vector µ(X t ) is being multiplied by the scalardt, yielding a new m-vector, and the m × n matrix σ(X t ) is begin multiplied by then-vector dW t , yielding a m-vector, so that we formally obtain⎛⎜⎝dX 1 t.dX m t⎞ ⎛⎟⎠ = ⎜⎝=⎛⎜⎝µ 1 (X t ).µ m (X t )⎞⎛⎟⎠ dt + ⎜⎝σ 11 (X t ) · · · σ 1n (X t ).µ 1 (X t ) dt + ∑ ni=1 σ 1i(X t ) dW i t.µ m (X t ) dt + ∑ ni=1 σ mi(X t ) dW i t.σ m1 (X t ) · · · σ mn (X t )⎞⎟⎠ ,⎞ ⎛⎟ ⎜⎠ ⎝dW 1 t.dW n t⎞⎟⎠

5.3 Monte Carlo evaluation of expectations 189in accordance with our first definition of the SDE.Analogously with the conventions for linear markets in Section 5.1, µ is called the driftcoefficient and σ is called the the volatility coefficient. We define a set-up as a filteredprobability space (Ω, F, P, F t ) satisfying the usual conditions and being endowed witha F t Brownian motion W . We say that the SDE has a weak solution if there for anyset-up exists a m-dimensional standard process X satisfying the SDE. We say that theSDE satisfies uniqueness in law if any two solutions on any two set-ups have the samedistribution. We say that the SDE satisfies pathwise uniqueness if any two solutionson the same set-up are indistinguishable.Furthermore, we say that the SDE has a strong solution if there exists a mappingF : R m × C([0, ∞), R n ) such that for any set-up with Brownian motion W and anyx ∈ R m , the process X x = F (x, W ) solves the SDE with initial condition x. We callF a strong solution of the SDE. If the SDE satisfies pathwise uniqueness, F (x, ·) isalmost surely unique when C([0, ∞)×R n ) is endowed with the Wiener measure. Whenthe SDE has a strong solution F , we can define the flow of the solution as the mappingφ : R m × C([0, ∞), R n ) × [0, ∞) → R m given by φ(x, w, t) = F (x, w) t . We say that theflow is differentiable if the mapping φ(·, w, t) is differentiable for all w and t.This concludes our review of fundamental notions for SDEs.5.3 Monte Carlo evaluation of expectationsAs we saw in the first section of this chapter, an arbitrage-free price of a contingentT -claim is given as a discounted expected value under the Q-measure. The problem ofevaluating prices can therefore be seen as a special case of the problem of evaluatingexpectations. As we shall see, the same is the case for the problem of evaluatingsensitivities of prices to changes in the model parameters. For this reason, we willnow review some basic Monte Carlo methods for evaluating expectations. Our basicresource is Glasserman (2003), in particular Chapter 4 of that work.Throughout the section, consider an integrable stochastic variable X. Our goal is tofind the expectation of X. The basis for the Monte Carlo methods of evalutatingexpectations is the strong law of large numbers, stating that if (X n ) is a sequence of

190 Mathematical Financeindependent variables with the same distribution as X, then1nn∑k=1X ka.s.−→ EX,This means that if we can draw a sequence of independent numbers sampled from thedistribution of X and take their average, the result should approach the expectation ofX as the number of samples tend to infinity. Furthermore, we can measure the speedof convergence by the variance,V(1n)n∑X k = 1 ∑nn 2 V X k = 1 n V X.k=1k=1In reality, we can of course not truly draw independent random numbers, we are onlyable to generate “pseudorandom” sequences, numbers which behave randomly in somesuitable sense. We will not discuss the implications of this observation, mostly becausethe pseudorandom number generators available today are of sufficiently high qualityto ensure that the topic is of little practical significance. For more information on thefundamentals of generating random numbers, see Chapter 2 of Glasserman (2003).The methods which we will concentrate upon are how the basic application of the lawof large numbers can be modified to speed up the convergence to the expectation, inthe sense of reducing the variance calculated above. The methods we will consider are,ordered by increased potential efficiency improvement:1. Antithetic sampling.2. Control variates.3. Stratified sampling.4. Importance sampling.Antithetic sampling. The idea behind antithetic sampling is to even out deviationsfrom the sample mean in the i.i.d. sequence. As before, we let (X n ) be an i.i.d.sequence with the same distribution as X. Let (X ′ n) be another sequence, and assumethat the sequence (X n , X ′ n) is also i.i.d. Assume furthermore that EX ′ n = EX n .Thus, the pairs are independent, and each marginal in any pair has the mean EX,

5.3 Monte Carlo evaluation of expectations 191it is however possible that X n and X n ′ are dependent, in fact such dependence is thevery idea of antithetic sampling. We then find( n) ()1 ∑ n∑X k + X k′ = 1 1n∑X k + 1 n∑X ′ a.s.k −→ EX.2n2 n nk=1k=1Thus, sampling from each sequence and averaging yields a consistent estimator of themean. If the samples from Xk ′ on average are larger than the mean when the samplesfrom X k are smaller than the mean and vice versa, we could hope that this would evenout deviations from the mean and improve convergence. To make this concrete, let X ′be such that (X, X ′ ) has the common distribution of (X n , X n). ′ We then obtain( (1 ∑ nVX k +2nk=1n∑k=1X ′ k))k=1= 1 ∑nn 2 Vk=1= 1 n 2 n ∑k=1k=1(Xk + Xk′ )214 (2V X k + 2Cov(X k , X ′ k))= 12n V X + 12n Cov(X, X′ ).The first term is the usual Monte Carlo estimator variance. Thus, we see that antitheticsampling can reduce variance if Cov(X, X ′ ) is negative, precisely corresponding to thevalues of X ′ k on average being larger than the mean when the values of X k are smallerthan the mean and vice versa.As a simple example of antithetic sampling, let us assume that we wanted to calculatethe second moment of the unit uniform distribution by antithetic sampling. We wouldneed to identify a two-dimensional distribution (X, X ′ ) such that X and X ′ both havethe means of the the second moment of the uniform distribution and are negativelycorrelated. A simple possibility would be to let U be uniformly distributed and defineX = U 2 and X ′ = (1 − U) 2 . X and X ′ are then both distributed as the squaresof uniform distributions. We would intuitively expect X and X ′ to be negativelycorrelated, since large values of X corresponds to large values of U, corresponding tosmall values of X ′ . In this case, we can check that this is actually true, sinceCov(X, X ′ ) = Cov(U 2 , (1 − U) 2 ) = − 112 .In practice, we would therefore consider a sequence (U n ) of uniform distributions andconstruct the expectation estimator( n)1 ∑ n∑Uk 2 + (1 − U k ) 2 ,2nk=1k=1

192 Mathematical Financewhich would be a consistent estimator of the mean, and its variance would be, sinceV U 2 = 4 45 , (1 42n 45 − 12) 1 . The is an improvement over the ordinary Monte Carlo estimatorof a factor 445 ( 445 − 1 12 )−1 = 16. This means that for any n, the antitheticestimator would have a standard deviation four times smaller than the ordinary estimator.Usually, the efficiency boost of antithetic sampling is not much larger thanthis, actually it is usually somewhat more modest. However, whenever we are consideringrandom variables based on distributions with some kind of symmetry, suchas the uniform distribution or the normal distribution, antithetic sampling is easy toimplement and almost always provides an improvement. There is therefore rarely anyreason not to use antithetic sampling in such situations.Control variates. Like antithetic sampling, the method of control variates operatesby attempting to balance samples which are below or above the true expectation bycorrelation considerations. Instead of adding an extra sequence of variables with thesame mean as the original sequence, however, the method of control variates adds anextra sequence of variables with some known mean which is not necessarily the sameas the mean of X.Therefore, let X ′ be some variable with known mean, possibly correlated with X.Suppose that the sequence (X n , X n) ′ is i.i.d. with the common distribution being thesame as that of (X, X ′ ). With b ∈ R, We can then consider the estimator1n(∑ n)n∑X k − b (X k ′ − EX ′ ) .k=1k=1The law of large numbers applies to show that this yields a consistent estimator. Theidea behind the estimator is that if X k and Xk ′ are correlated, we can choose b suchthat the deviations from the mean of X k are evened out by the deviations from themean of Xk ′ . In the case of positive correlation, this would mean choosing a positiveb, since large values of X k then would be offset by large values of Xk ′ and vice versa.Likewise, in the case of negative correlation, we would expect that choosing a negativeb would yield the best results. We call Xk ′ the control variate.We will now analyze the variance of the control variate estimator. We find( ( n))1 ∑n∑V X k − b (X k ′ − EX ′ ) = 1 ∑nnn 2 V (X k − b(X k ′ − EX ′ ))k=1k=1k=1= 1 n (V X k + b 2 V (X ′ k) − 2bCov(X k , X ′ k))= 1 n V X + 1 n(b 2 V X ′ − 2bCov(X, X ′ ) ) .

5.3 Monte Carlo evaluation of expectations 193The first term is the variance of the ordinary Monte Carlo estimator. If the controlvariate estimator is to be useful, the second term must be negative, corresponding tob 2 V X ′ −2bCov(X, X ′ ) < 0. To find out when we can make sure this is the case, we firstidentify the optimal value of b. This is the value minimizing b 2 V X ′ − 2bCov(X, X ′ )over b. The formula for the minimum of a quadratic polynomial yields the optimalvalue b ∗ = Cov(X,X′ )V X. The corresponding variance change is′(b ∗ ) 2 V X ′ − 2b ∗ Cov(X, X ′ ) = − Cov(X, X′ ) 2V X ′ .Since this is always negative, we conclude that, disregarding considerations of computingtime, control variates can only improve convergence, and there is an improvementwhenever X and X ′ are correlated. The factor of improvement of the control variateestimator to the ordinary estimator isV X1=V X − Cov(X,X′ ) 21 − Corr(X, X ′ ) 2 .V X ′This means that if, for example, the correlation is 1 2, the variance of the controlvariate estimator is a factor of 4 3larger than the variance of the ordinary Monte Carloestimator. If the correlation is 3 4 , it is a factor of 16 7larger and so on. The improvementgrows drastically as the correlation tends to one, with the limit corresponding to thedegenerated case where X ′ has the distribution of X, which is of course absurd sinceEX then would be known, making the estimation unnecessary.As an example, we can again consider the problem of numerically evaluating the secondmoment of the uniform distribution. As we have seen, to find an effective controlvariate, we need to identify a variable which is highly correlated with the variableswe are using to estimate the second moment, and whose mean we know. Thus, let Ube uniformly distributed and let X = U 2 . Assuming that we know that the mean ofthe uniform distribution is 1 2 , we can use the control variate X′ = U. Obviously, Xand X ′ should be quite highly correlated. Note that unlike the method of antitheticsampling, it does not matter whether the correlation is positive or negative, this istaken care of in the choice of coefficient. We find that the correlation isCorr(X, X ′ ) = Cov(U 2 , U)√V U2 √ V U = √4548 ,which is pretty close to one, leading to an reduction in variance of the estimator to1the ordinary estimator of a factor = 16. The optimal coefficient is1− 4548Cov(X, X ′ )V X ′ = Cov(U 2 , U)= 1.V U

194 Mathematical FinanceThe corresponding control variate estimator is, letting (U n ) be a sequence of i.i.d.(∑uniforms, 1 nn k=1 U k 2 − ∑ n(k=1 Uk − 2)) 1 . As expected, very large values of Uk correspondto shooting over the true mean, and this is offset by the control variate. Sincethe control variate method in this case does not entail much extra calculation, wecan reasonably say that the associated computational time is the same as that ofthe ordinary Monte Carlo estimator. Therefore, we appreciate that the control variatemethod provides an estimator with one fourth of the standard deviation of theordinary estimator with the same amount of computing time.In practice, the optimal coefficient cannot be calculated explicitly. Instead, one canapply the Monte Carlo simulations to estimate it, usingˆb∗ =∑ nk=1 (X k − X)(X ′ k − X′ )∑ nk=1 (X′ k − X′ ) 2 .The extra computational effort required from this is usually more than offset by thereduction in standard deviation.Stratified sampling. Stratified sampling is a sampling method where the samplespace of the distribution being sampled is partitioned and the number of samples fromeach part of the partition is being controlled. In this way, the deviation from the meancannot go completely awry.We begin by introducing a stratification variable, Y . Let A 1 , . . . , A m be disjoint subsetsof R such that P (Y ∈ ∪ m i=1 A i) = 1 and P (Y ∈ A i ) > 0 for all i. We call these sets thestrata. The idea is that instead of sampling from the distribution of X to obtain themean of X, we will sample from the distribution of X given Y ∈ A i .Let X i have the distribution of X given Y ∈ A i . We then find, putting p i = P (Y ∈ A i ),m∑m∑( )X1(Y ∈Ai)EX = EX1 (Y ∈Ai ) = p i E=p ii=1i=1m∑p i EX i ,so if we can sample from each of the conditional distributions, we can make MonteCarlo estimates of each term in the sum above and thereby obtain an estimate forEX. To make this concrete, let for each i (X ik ) be an i.i.d. sequence with thecommon distribution being the same as that of X i . All these variables are assumedindependent. Fix some n. We are free to choose how many samples to use from eachstrata. We will restrict ourselves to what is known as proportional allocation, meaningthat we choose to draw a fraction p i of samples from the i’th stratum. Assume fori=1

5.3 Monte Carlo evaluation of expectations 195simplicity that np i is an integer, we can then put n i = np i , obtain n = ∑ mi=1 n i anddefine the stratified Monte Carlo estimatorBy the law of large numbers,1nm∑ ∑n im∑X ik =i=1 k=11nm∑ ∑n iX ik .i=1 k=1∑npiiX ikni=1 ik=1a.s.−→so the estimator is consistent. Its variance is()1m∑ ∑n iVX ik = 1 ∑ m∑n inn 2 V X ik = 1 ni=1 k=1i=1 k=1m∑p i EX i = EXi=1m∑p i V X i .To compare this with the variance of the ordinary Monte Carlo estimator, define U byletting U = i if Y ∈ A i . Then, the distribution of Y given U = i is the distribution ofX i , and for the distribution of U we find P (U = i) = P (Y ∈ A i ) = p i . All in all, wemay conclude1ni=1m∑p i V X i = 1 n EV (X|U) = 1 n V X − 1 n V E(X|U),i=1where the first term is the variance of the ordinary Monte Carlo estimator. Sincethe second term is a negative scalar times a variance, it is always negative. Therefore,disregarding questions of computation time, the convergence of the stratified estimatoris always at least as fast as that of the ordinary estimator.Let us review the components of the stratified sampling technique. We start out withX, the distribution whose mean we wish to identify. We then consider a stratificationvariable Y and strata A 1 , . . . , A m . It is the simultaneous distribution of (X, Y ) combinedwith the strata which determine how our stratified estimator will function. Wedefine i.i.d. sequences X ik such that X ik has the distribution of X given Y ∈ A i . Theestimator 1 ∑ m ∑ nin i=1 k=1 X ik is then consistent and has superior variance compared tothe ordinary Monte Carlo estimator.In practice, the critical part of this procedure is to certify that it is possible andcomputationally tractable to sample from the distribution of X given Y ∈ A i . Thisis often accomplished by ensuring that X = t(Y ) for some transformation t. In thatcase, the distribution of X given Y ∈ A i is just the distribution of Y given Y ∈ A itransformed with t. If Y is a distribution such that it is easy to sample from Y

196 Mathematical Financegiven Y ∈ A i , we have obtained a simple procedure for sampling from the conditionaldistribution. This is the case if, say, Y is a distribution with an easily computablequantile function and A i is an interval.To see an example of how stratified sampling can be used, we will use it to numericallycalculate the second moment of the uniform distribution. Let U be uniformlydistributed, we then wish to identify the mean of U 2 . Fix m ∈ N, we will use U asour stratification variable with the strata A i = ( i−1m , i m ]. We then find p i = 1 m andn i = n m . We need to identify the distribution of U 2 given U ∈ A i . We know thatthe distribution of U given U ∈ A i is the uniform distribution on A i . Therefore, U 2given U ∈ A i has the same distribution as ( i−1m+ U m )2 . Letting U ik be unit uniformlydistributed i.i.d. sequences, we can write our stratified estimator asIts variance is1n= 1 n= 1 n= 1 n= 1 nm∑i=1m∑i=1m∑i=1m∑i=1m∑i=11nm∑i=1 k=1nm∑( i − 1m(1 i − 1m V m+ U ) 2im1m 5 V (i − 1 + U i) 2+ U ) 2ik.m1m 5 V ((i − 1)2 + 2(i − 1)U i + U 2 i )1 (4(i − 1) 2m 5 V U i + V Ui 2 + 4(i − 1)Cov(U i , Ui 2 ) )(1 (i − 1)2m 5 + 4 3 45 + i − 13= 1 (1 (m − 1)m(2m − 1)n m 5 18+ 4m45))(m − 1)(m − 2)+ ,3which is of order O(m −2 ) when m tends to infinity. This basically shows that we intheory can obtain as small a variance as we desire by picking as many strata as possible.In reality, of course, it is insensible to pick more strata than observations. The moralof the example, however, is the correct one: by using a large number of strata, a verydramatic reduction of variance is sometimes possible. An example of how stratifiedsampling can improve results can be seen in Figure 5.1, where we compare samplingfrom squared uniforms with and without stratification. We see that the histogramon the right, using stratified sampling, is considerably closer to the true form of thedistribution.

5.3 Monte Carlo evaluation of expectations 197Frequency0 1 2 3 4Frequency0 1 2 3 40.0 0.2 0.4 0.6 0.8 1.0Value0.0 0.2 0.4 0.6 0.8 1.0ValueFigure 5.1: Comparison of samples from a squared uniform distribution with andwithout stratified sampling. On the left, 500 samples without stratification. On theright, 500 samples with 100 strata.The ease of implementation and potentially large improvements make stratified samplingan very useful method for improving convergence.Importance sampling. Importance sampling is arguably the method described herewhich has the largest potential impact. Its usefulness varies greatly from situationto situation, and if used carelessly, it can significantly reduce the effectivity of anestimator. On the other hand, in situations where no other method is applicable, itsometimes can be applied to obtain immense improvements.The underlying idea of importance sampling is very simple. It is most naturally describedin the context where X = h(X ′ ) for some variable X ′ and some mapping hwith sufficient integrability conditions. Let µ be the distribution of X ′ . Let Y ′ besome other variable with distribution ν and assume that µ ≪ ν. We can then use theRadon-Nikodym theorem to write∫∫Eh(X ′ ) = h(x) dµ(x) =h(x) dµ(dν dν(x) = E h(Y ′ ) dµ )dν (Y ′ ) .Letting Y n ′ be a i.i.d. sequence with common distibution being the same as that of Y ′ ,we can then define the importance sampling estimator as1n∑h(Y ′nk) dµdν (Y k).′k=1The law of large numbers yields that this is a consistent estimator of Eh(X ′ ).Its

198 Mathematical Financevariance can be larger or smaller than that of the ordinary Monte Carlo estimatordepending on the choice of ν. Results on the effectiveness of the importance samplingestimator as a function of the choice of ν can only be obtained in specialised situations.Furthermore, the objective of the importance sampling method will in fact often notbe to reduce variance, but rather to change the shape of a distribution when it issomehow suboptimal for the purposes of Monte Carlo estimation. Is is difficult todescribe quantitatively when this is the case, we will instead give a small example.Let X ′ be the standard exponential distribution and consider the problem of estimatingP (X ′ > α) for some large α. This means considering the transformation h given byh(x) = 1 (α,∞) (x). Let p = P (X > α), the variance of h(X ′ ) is p(1−p), which is prettysmall. In theory, then, the ordinary Monte Carlo estimator should yield relativelyeffective estimates. However, this is not actually the case.To see why the ordinary estimator often will underestimate the true probability, wecannot rely only on moment considerations, we need to analyze the distribution ofthe estimator in more detail. To this end, note that 1 ∑ nn k=1 h(X′ k) is binomially distributedwith length n, probability parameter p and scale 1 n. A fundamental problemwith this is that the binomial distribution is discrete, this immediately puts an upperlimit to the accuracy of the estimator. Also, the estimator will often underestimatethe true probability. To see how this is the case, simply note that the probability ofthe estimator being zero is (1 − p) n . When p is small compared to n, this can be verylarge. In the case α = 10, for example, we obtain p = e −10 , (1 − p) 10000 ≈ 63% and(1 − p) 100000 ≈ 6.3%. This means that with 10000 samples, our estimator will yieldzero in 63% of our trials. Using ten times as many simulations yields zero only 6.3%percent of the time, but the distribution will still very often be far from the correctprobability, which in this case is e −10 ≈ 0.0000453.We can use importance sampling to improve the efficiency of our estimator. We letthe importance sampling measure ν be the exponential distribution with scale λ. Wethen havedµdν (x) =e−x1λ e− x λ( (= λ exp −x 1 − 1 )),λand with (Y n) ′ being an i.i.d. sequence with distribution ν, the importance samplingestimator is1n∑( (h(Y ′nk)λ exp −Y k′ 1 − 1 )).λk=1In Figure 5.2, we have demonstrated the effect of using the importance sampling estimator.We use α = 10 and compare the histograms of 1000 independent ordinary

5.4 Methods for evaluating sensitivities 199Monte Carlo estimators with 1000 independent importance sampling estimators, lettingλ = 10 and basing the estimators on 1000 samples.Frequency0 200 400 600 800Frequency0 20 40 60 80 100 120 140Frequency0 50 100 1500e+00 2e−04 4e−04 6e−04 8e−04 1e−030e+00 2e−04 4e−04 6e−04 8e−04 1e−030e+00 2e−05 4e−05 6e−05 8e−05 1e−04ValueValueValueFigure 5.2: Comparison of estimation of exponential probabilities with and withoutimportance sampling. From left to right, histogram of 1000 ordinary Monte Carloestimators, histogram of 1000 importance sampling estimators and finally, close-uphistogram of 1000 importance sampling estimators.Obviously, the ordinary Monte Carlo estimator cannot get very close to the true valueof p because of its discreteness. As expected, it hits zero most of the time, with a1few samples hitting1000. On the other hand, the importance sampling estimator cantake any value in [0, nc) for some c > 0, removing any limits on the accuracy arisingfrom discreteness. We see that even with only the 1000 samples which were completelyuseless in the ordinary Monte Carlo case, the importance sampling estimator hits thetrue probability very well most of the time. Furthermore, the computational overheadarising from the importance sampling technique is minimal. We conclude that withvery little effort, we have improved the ordinary estimator by an extraordinary factor.5.4 Methods for evaluating sensitivitiesNext, we turn our attention to methods specifically designed to evalutate sensitivitiessuch as the delta or gamma of a claim. We will cast our discussion in a generalframework, which will then be specialised to financial cases in the following sections.We assume that we have a family of integrable variables X(θ) indexed by an opensubset Θ of the real line. Our object is to understand how to estimate ∆ θ = d dθ EX(θ),assuming that the derivative exists. We will consider five methods, being:

200 Mathematical Finance1. Finite difference.2. Pathwise differentiation.3. Likelihood ratio.4. Malliavin weights.5. Localisation.These methods in general transform the problem of estimating a derivative into estimatingone or more expectations. These expectations can then be evaluated by themethods described in Section 5.3. In the following we always assume that the mappingθ ↦→ EX(θ) is differentiable.Finite difference. The finite difference method aims to calculate the sensitivity byapproximating the derivative by a finite difference quotient and estimating this byMonte Carlo. Define α(θ) = EX(θ). Fixing θ and leting h > 0, our approximation isdefined byα(θ + h) − α(θ − h)ˆ∆ θ = .2hIntroducing i.i.d. variables (X n (θ + h), X n (θ − h)) such that X n (θ + h) and X n (θ − h)has the distributions of X(θ+h) and X(θ−h), respectively, but possibly are correlated,we can define the finite difference estimator asX n (θ + h) − X n (θ − h).2hThe law of large numbers yields X n(θ+h)−X n (θ−h)2ha.s.−→ ˆ∆ θ , so the estimator is in generalbiased. To analyze the bias, note that if α is C 2 , we can form second-order expansionsof α at θ, yieldingα(θ + h) = α(θ) + α ′ (θ)h + 1 2 α′′ (θ)h 2 + o(h 2 )α(θ − h) = α(θ) − α ′ (θ)h + 1 2 α′′ (θ)h 2 + o(h 2 ),so that in this case, the bias has the formˆ∆ θ − α ′ (θ) =α(θ + h) − α(θ − h)2hIf α is C 3 , we can obtain the third-order expansions− α ′ (θ) = o(h).α(θ + h) = α(θ) + α ′ (θ)h + 1 2 α′′ (θ)h 2 + 1 6 α′′′ (θ)h 3 + o(h 3 )α(θ − h) = α(θ) − α ′ (θ)h + 1 2 α′′ (θ)h 2 − 1 6 α′′′ (θ)h 3 + o(h 3 ),

5.4 Methods for evaluating sensitivities 201which yields the bias 1 6 α′′′ (θ)h 2 + o(h 2 ), which is not only o(h) but is O(h 2 ).Conditional on a level of smoothness of α, then, the finite difference estimator is at leastasymptotically unbiased. Letting (X ′ (θ + h), X ′ (θ − h)) have the common distributionof (X n (θ + h), X n (θ − h)), the variance of the estimator is( )Xn (θ + h) − X n (θ − h)V2h==( n)14h 2 n 2 V ∑X k (θ + h) − X k (θ − h)k=114h 2 n V (X′ (θ + h) − X ′ (θ − h)).We see that the variance is highly dependent both on h and on the dependence structurebetween X ′ (θ + h) and X ′ (θ − h). Both of these are under our control, and wecan therefore seek to choose them in a manner which reduces the mean square erroras much as possible. This will provide a reasonable tradeoff between reducing biasand reducing variance. The context which we are working in at present is too generalfor us to be able to make any optimal choice of the dependence between X ′ (θ + h)and X ′ (θ − h), so we will content ourselves with making some assumptions about theasymptotic behaviour of the bias and variance and examining how best to choose h asa function of n from this.We will assume thatˆ∆ θ − α ′ (θ) = O(h 2 )V (X ′ (θ + h) − X ′ (θ − h)) = O(h β ),for some β > 0. This corresponds to situations often encountered in practice. We thendefine h n = cn −γ and want to identify the rate of decrease γ yielding the asymptoticallyminimal mean square error. The mean square error isMSE( ˆ∆ n ) = V ( ˆ∆ n ) + (E ˆ∆ n − θ) 2=14h 2 nn V (X′ (θ + h n ) − X ′ (θ − h n )) + O(h 2 n) 2=14h 2 nn O(hβ n) + O(h 4 n)=14c 2 n 1−2γ O(n−βγ ) + O(n −4γ )= O(n 2γ−1 )O(n −βγ ) + O(n −4γ )= O(n γ(2−β)−1 ) + O(n −4γ )

202 Mathematical FinanceThis is O(n δ(γ) ), where δ(γ) = max{γ(2 − β) − 1, −4γ}. From this we see that witha small positive γ, we obtain δ(γ) < 0. This implies that n −δ(γ) MSE( ˆ∆ n ) is boundedas n tends to infinity. Since n −δ(γ) tends to infinity, we conclude that in this case,MSE( ˆ∆ n ) tends to zero. In other words, if we decrease h as n tends to zero, themean square error tends to zero and we obtain a consistent estimator. This is a veryreasonable conclusion.However, if β < 2, choosing a γ which is too large, corresponding to a very fastdecrease in the step size, can end up yielding δ(γ) > 0, corresponding to increasingmean square error. We see that choosing γ properly is paramount to obtaining effectivefinite difference estimates. The condition β < 2 corresponds to the situation wherethe decrease in V (X ′ (θ + h) − X ′ (θ − h)) cannot keep up with the factor 1 hin the2expression for the total variance of ˆ∆ n , and therefore large γ can cause an explosionin variance. The optimal choice of γ is found by minimizing δ over γ, leading to thefastest decrease of the mean squared error.To this end, note that a negative γ always yields δ(γ) > 0, so this is never a goodchoice. It will therefore suffice to optimize over positive γ. We split up according towhether β < 2, β = 2 or β > 2. If β < 2, δ(γ) is the maximum of a line with increasingslope and a line with decreasing slope. The minimum is therefore where the two linesintersect, meaning the solution to γ(2 − β) − 1 = −4γ, which is γ = 16−β. If β = 2, theminimum is obtained for any γ ≥ 1 4. The optimal choice of γ would in this case dependon the explicit form of the mean squared error. If β > 2, δ is downwards unboundedand γ should be chosen as large as possible. This latter case, however, probably willnot happen in reality, at least not in any of the practial cases we consider.Note that our analysis is somewhat different from the one found in Section 7.1 ofGlasserman (2003), and the case β > 2 does not correspond to any of the casesconsidered there. This is because Glasserman (2003) always assumes that the totalvariance of the estimator tends to zero as h tends to zero. In a sense, the parametrizationof the orders of decrease is different. The results, however, are equivalent.To apply the finite difference method, then, very little actual implementation work isneeded, but it may be necessary to spend some time figuring out the best selection ofh.We now consider two examples of application of the finite difference method. First, letU(θ) have the normal distribution with unit variance and mean θ, assume that we are

5.4 Methods for evaluating sensitivities 203interested in estimating d dθ EU(θ)2 . If we let U ′ (θ + h) and U ′ (θ − h) be independentnormal variables with unit variance and means θ + h and θ − h, respectively, we obtainV ( U ′ (θ + h) 2 − U ′ (θ − h) 2) = V U ′ (θ + h) 2 + V U ′ (θ − h) 2= 2(1 + 2(θ + h) 2 ) + 2(1 + 2(θ − h) 2 )= 4 + 4((θ + h) 2 + (θ − h) 2 )= 4 + 8θ 2 + 8h 2 ,which is O(1) as h tends to zero. We have used that U ′ (θ + h) 2 has a noncentral χ 2distribution to obtain the variances. This corresponds to a β value of zero, and wewould therefore expect that the optimal choice of step size is h n = cn − 1 6 for some c.This would yields a total variance of4 + 8θ 2 + 8c 2 n − 1 3,4c 2 n 1− 1 3which is of order o(n). To find the corresponding bias of the estimator, note thatputting α(θ) = EU(θ) 2 = 1 + θ 2 , we find α ′ (θ) = 2θ. The bias is then1 + (θ + h) 2 − (1 + (θ − h) 2 )2h− 2θ = 0,so the estimator is unbiased. On the other hand, if we let U be a standard normal anddefine U ′ (θ + h) = U + θ + h and U ′ (θ − h) = U + θ − h, we findV ( U ′ (θ + h) 2 − U ′ (θ − h) 2) = V ( U 2 + 2(θ + h)U + h 2 − (U 2 + 2(θ − h)U + h 2 ) )= V (4hU)= 16h 2 ,which is O(h 2 ). Using this simulation scheme then yields β = 2 and a reasonablechoice is h n = cn −γ for any γ ≥ 1 4. The total variance becomes16c 2 n −2γ4c 2 n 1−2γ = 4 n ,which, as expected, is asymptotically independent of γ (in fact, completely independent).If the estimator were biased, we could have used our freedom in choosing γ tominimize the bias.Pathwise differentiation. In the finite difference method, we considered a finitedifference approximation to d dθEX(θ) and let the parameter increment tend to zeroat the same time as we let the number of samples tend to infinity. In the method ofpathwise differentiation, we instead simply exchange differentiation and mean and use

204 Mathematical Financeordinary Monte Carlo to estimate the result. Thus, the basic operation we desire tomake isddθ EX(θ) = E d dθ X(θ).While the left-hand side is only dependent on each of the marginal distributions X(θ),the right-hand side is dependent on the distribution of the ensemble (X(θ)) θ∈Θ . Forthe interchange to make sense at all, we therefore assume that we are working ina context where all of the variables X θ are defined on the same probability space.Furthermore, it is of course necessary that the variable d dθX(θ) is almost surely welldefinedand integrable, so we need to assume that X θ is almost surely differentiable inθ for any θ ∈ Θ. Finally, we need to make sure that the exchange of integration anddifferentiation is allowed.If all this is the case, we can let X n(θ) ′ be an i.i.d. sequence with common distributiondbeing the same as the distribution ofdθX(θ) and define the pathwise differentiationestimator1n∑X ′nk(θ),k=1which by the law of large numbers and our assumptions satisfies1nn∑k=1X ′ k(θ) a.s.−→ E d dθ X(θ) = d dθ EX(θ),so the pathwise differentiation estimator, when it exists, is always consistent. However,we have no guarantees that the variance of the pathwise estimator is superior to thatof the finite difference estimator. Two definite benefits of the pathwise differentiationmethod over the finite difference method, though, are its unbiasedness and that we nolonger need to worry about the optimal parameter increment in the difference quotient.Other than that, we have no guarantees for improvement. General experience statesthat the pathwise differentiation estimator rarely is worse than the finite differenceestimator, but also rarely yields more than moderate improvements in variance.We will now consider sufficient conditions for the existence and consistency of thepathwise differentiation estimator. For clarity, we reiterate that we at all times assume1. The variables X θ are all defined on the same probability space.2. For each θ, X(θ) is almost surely differentiable.3. The variable d dθX(θ) is integrable.

5.4 Methods for evaluating sensitivities 205A sufficient condition for the interchange of differentiation and integration is thenthat θ is if the family ( X(θ+h)−X(θ)h) is uniformly integrable over h in some puncturedneighborhood of zero: Since the variables converge almost surely to d dθX(θ), we alsohave convergence in probability, and this combined with uniform integrability yieldsconvergence in mean and therefore convergence of the means, that is,( ) ()dX(θ + h) − X(θ)EX(θ) = limdθ E X(θ + h) − X(θ)= E lim= E dh→0 hh→0 hdθ X(θ),as desired. We now claim that the following is a sufficient condition for the interchangeof differentiation and integration:• V (X(θ + h) − X(θ)) = O(h 2 ).To see this, note that under this condition,E(X(θ + h) − X(θ)) 2 = V (X(θ + h) − X(θ)) + E 2 (X(θ + h) − X(θ))= O(h 2 ) + h 2 ( EX(θ + h) − EX(θ)h= O(h 2 ),so the family X(θ+h)−X(θ)his bounded in L 2 over a punctured neighborhood of zero. Inparticular, it is uniformly integrable, so the interchange is allowed by what we alreadyhave shown. Another sufficient condition is:) 2• For θ ∈ Θ, there is an integrable variable κ θ such that |X(θ + h) − X(θ)| ≤ κ θ |h|for all h in a punctured neighborhood of zero.In order the realize this, merely note that in this case, κ θ is an integrable bound for|X(θ+h)−X(θ)|h, and therefore the dominated convergence theorem yields the result.We now turn to a concrete example illustrating the use of the pathwise differentiationmethod. We consider the same situation as in the subsection on the finite differencemethod, estimating d dθ EU(θ)2 where U(θ) is normally distributed with unit varianceand mean θ. To use the pathwise method, we first need to make sure that all of thevariables are defined on the same probability space. This is easily obtained by lettingU be some variable with the standard normal distribution and defining U(θ) = U + θ.Clearly, then, θ ↦→ (U + θ) 2 is always everywhere differentiable, and the derivative is

206 Mathematical Financeddθ (U + θ)2 = 2(U + θ), which is integrable. We earlier foundV (U(θ + h) 2 − U(θ) 2 ) = 4h 2 = O(h 2 ),so by our earlier results, the interchange of differentiation and integration is allowed,and we concludeddθ EU(θ)2 = E d dθ U(θ)2 = E2(U + θ).Now, we can of course evalutate this expectation directly, but for the sake of comparingwith the finite difference method, let us consider what would happen if we used a MonteCarlo estimator. The variance based on n samples would then be 1 n V (2U + 2θ) = 4 n ,which is the same as the finite difference estimator.Likelihood ratio. The likelihood ratio method, like the pathwise differentiationmethod, seeks to evaluate the sensitivity by interchanging differentiation and integration.But instead of differentiating the variables themselves, the likelihood ratiomethod differentiates their densities. Since densities usually in most cases are relativelysmooth, while the stochastic processes often, in particular in the case of optionvaluation, have discontinuities or breaks, this means that the likelihood ratio methodin many cases applies when the pathwise differentiation method does not.To set the stage, assume that X(θ) = f(X ′ (θ)) for some X ′ (θ) with density g θ , wheref : R → R is some measurable mapping. This is the formulation best suited to laterapplications. Our goal is then to evaluate d dθ Ef(X′ (θ)). The basis of the likelihoodratio method is the observation that if the interchange is allowed and the denominatorsare nonzero, we haveddθ Ef(X′ (θ)) = d ∫f(x)g θ (x) dxdθ∫= f(x) d dθ g θ(x) dx=∫f(x) g′ θ (x)g θ (x) g θ(x) dx= Ef(X ′ (θ)) g′ θ (X′ (θ))g θ (X ′ (θ)) .While the pathwise method rewrote the sensitivity as an expectation of a somewhatdifferent variable, the likelihood ratio method rewrites the sensitivity as a new transformationof the old variable.As with the pathwise method, there is no guarantee that the likelihood ratio methodyields better results than the finite difference estimator. In fact, it is easy to find

5.4 Methods for evaluating sensitivities 207cases where it yields substantially worse results than finite difference, and we shall seeexamples of this later. As a rule of thumb, if the mapping f has discontinuities, thelikelihood ratio has a good chance of improving convergence, otherwise it may easilyworsen convergence. In this sense, the likelihood method complements the pathwisemethod: the pathwise method is usually useful in the presence of continuity, and thelikelihood ratio is usually useful in the presence of discontinuities. A limitation of thelikelihood ratio method is that it is necessary to know the density of the variables andtheir derivatives, as this is rarely the case in practice. However, the scheme can beextended to cover very general cases by approximation arguments, see Section 7.3 ofGlasserman (2003).There are no criteria in simple terms for when the interchange of differentiation andintegration of the likelihood ratio method is allowed. However, general criteria can beexpressed in the case where the family X ′ (θ) is an exponential family. See for examplethe comments to Lemma 1.3 of Jensen (1992), page 7.Malliavin weights. The method of Malliavin weights, much as the likelihood method,computes sensitivities by transforming the derivative of the expectation into anotherexpectation. The method does not work in general cases, but is specifically suited tothe context where the sensitivity to be calculated is with respect to a transformationof the solution to an SDE. Specifically, consider a n-dimensional Brownian motion Wand assume that the SDEdX t = µ(X t ) dt + σ(X t ) dW thas a strong solution F , such that for each x ∈ R n , X x = F (x, W ) is a solution to theabove SDE with initial condition x ∈ R n . We further assume that the correspondingflow is differentiable and let Y x be the Jacobian of X x . Furthermore, let T > 0 besome constant and let f : R n → R be some mapping such that Ef(XT x)2 is finite.The basic result which we will need to derive the Malliavin estimator is that puttingu(x) = Ef(XT x ), it holds under suitable regularity conditions that(∇u(x) = E f(XT x ) 1 ∫ )T(σ −1 (Xt x )Yt x ) t dW tTThis result is Proposition 3.2 of Fournié et al. (1999). Here, σ −1 denotes matrixinverse and not the inverse mapping. We are not able to prove this with the theorythat we have developed. The proof exploits several of the results from Section 4.5,among others the duality formula. It also uses the Diffeomorphism Theorem andcertain criteria for uniform integrability from the theory of SDEs. See the notes forfurther details and references.0

208 Mathematical FinanceWe will now see how this result specialises to the case of one-dimensional financialmarkets. First consider the case where the financial instrument is given on the formdS t = rS t dt + σ(S t )S t dW t for some positive measurable mapping σ. Let f be suchthat Ef(S T ) 2 is finite, we will find the delta of the T -claim f(S T ). We assume thatthe short rate is zero, our results will then obviously immediately extend to the caseof deterministic short rate. Assuming that the result from Fournié et al. (1999) can∂be applied, we can use the fact that∂S 0S T = S TS0to immediately conclude(∂Ef(S T ) = E f(S T ) 1 ∫ T∂S 0 T 0(= Ef(S T ) 1T S 0∫ T1σ(S T )S TS TS 0dW t)01σ(S T ) dW tThis is the basis for the Malliavin weight estimator of the delta in the case where thevolatility is of the local form. This applies, for example, to the Black-Scholes model.Next, we consider a market of the form).dS t = rS t dt + σ t S t√1 − ρ2 dW 1 t + σ t S t ρ dW 2 tdσ t = α(σ t ) dt + β(σ t ) dW 2 t ,with constant initial conditions S 0 and σ 0 , where we assume that α and β are differentiableand ρ ∈ (−1, 1). We further assume that β is positive. This covers for examplethe Heston model. As before, consider f such that Ef(S T ) 2 is finite, we desire to find∂the delta of the T -claim f(S T ), that is,∂S 0Ef(S T ). To do so, we defineµ(x) =(rx 1α(x 2 )and put X t = (S t , σ t ). We then obtain)and σ(x) =(√ )x 2 x 1 1 − ρ2x 2 x 1 ρ,0 β(x 2 )dX 1 (t) = µ 1 (X t ) dt + σ 11 (X t ) dW 1 t + σ 12 (X t ) dW 2dX 2 (t) = µ 2 (X t ) dt + σ 21 (X t ) dW 1 t + σ 22 (X t ) dW 2 t∂with the initial conditions X 0 = (S 0 , σ 0 ). Note that∂S 0Ef(S T ) =∂∂X 1 (0) Ef(X 1(T )).This observation leads us to believe that we may be able to use the result from Fourniéet al. (1999) to obtain an expression for the sensitivity. Therefore, we assume thatthere is a strong solution to this SDE, yielding a differentiable flow, and we assumethat the results of Fournié et al. (1999) applies to our situation. Let Y be the first

5.4 Methods for evaluating sensitivities 209variation process of X under the initial condition X 0 . We find⎛⎞√1ρ(−√ σ −1 (X t )Y t = ⎝ Xt 2X1 t 1−ρ 2 1−ρ 2 β(Xt 2)⎠=⎛⎝X 2 t X1 t01β(X 2 t )Y 11t√ − ρY 21t1−ρ 2Yt 11 Yt12Yt 21 Yt22Y√ √ 12t− ρY√ 22t1−ρ 2 β(Xt 2) Xt 2X1 t 1−ρ 2 1−ρ 2 β(Xt 2)Y 21tY 22β(Xt 2) tβ(Xt 2))⎞⎠ .Because X 2 does not depend on X0 1 , Yt 21 = ∂ X 2 ∂X0 1 t = 0. And since X 1 is given as thesolution to an exponential SDE, Y 11 = Xt 1 = 1 X 1 X0 1 t . Therefore, we can reducethe above to⎛σ −1 (X t )Y t = ⎝1X 2 t X1 0√1−ρ 20∂∂X 1 0X 2 t X1 tY 12t√ − ρY 22t1−ρ 2Y 22tβ(X 2 t )√1−ρ 2 β(X 2 t )⎞⎠ .Transposing this, the result from Fournié et al.(∂Ef(XT 1 ) = E f(XT 1 ) 1 ∫ TT∂X 1 00(1999) yields)1√Xt 2 X dW 1 01 1 − ρ2t .Collecting our results and substituting S and σ for X 1 and X 2 , we may conclude(∂Ef(S T ) = E f(S T ) 1 ∫ )T1√∂S 0 S 0 T σ dW 1 t 1 − ρ2t ,and this is the Malliavin estimator for the delta. Note that this is not the sameestimator as is obtained in Benhamou (2001). We will return to this point later, whenconsidering the practical implementation of the Malliavin estimator in the Hestonmodel.0Localisation. The method of localisation is more of a shrewd observation than anactual method. It can in principle be used in conjunction with any of the MonteCarlo methods we have described, but its primary usefulness for us will be togetherwith the likelihood ratio and Malliavin methods. We show how the method applies tothe likelihood ratio method, the application to the Malliavin method is similar. Wetherefore consider the same situation as in our discussion of the likelihood method,where we put X(θ) = f(X ′ (θ)) and desire to evaluate d dθ EX(θ). Letting g θ be thedensity of X ′ (θ) and putting α θ (x) = g′ θ g θ (x), the likelihood ratio method yields theequalityddθ EX(θ) = Ef(X′ (θ))α θ (X ′ (θ)).

210 Mathematical FinanceNow, if α θ has a large variability and the supports of f and α θ have a large commonsupport, the factor α θ (X ′ (θ)) could be instrumental in making the variance of estimatorsbased upon the above rather large. Such detrimental effects are actually observedin reality, which means that the removal of the derivative in such cases have a verylarge price, and ordinary finite difference methods may be more effective.One way to avoid such enlargements of variance is to make the simple observation thatfor any sufficiently integrable mapping h, we haveddθ EX(θ) = d dθ Eh(X′ (θ)) + d dθ E(f − h)(X′ (θ))= Eh(X ′ (θ))α θ (X ′ (θ)) + d dθ E(f − h)(X′ (θ)).Here, we have split the mapping f into two parts, h and f − h, and have applied thelikelihood ratio method only to the first part. The second term can be evaluated usingother methods, such as the finite difference method or the pathwise method. If we canchoose h such that the variance of h(X ′ (θ))α θ (X ′ (θ)) is small, then we have obtainedan improvement over the usual likelihood ratio estimator provided that the secondterm does not add too much variance. One way of obtaining this reduced varianceis to make sure that the h has a small support. The choice of h, however, should bemade such that the usefulness of the likelihood ratio method is preserved and suchthat the remained term can be calculated with other methods. In other words, we wewould like the h to have small support and f − h to have reasonable smoothness. Oneexample of this is when working with the digital option, where f − h can be chosenas a smoothened version of the digital payoff, and h is then a kind of small bumpfunction with a jump. This enables one to use the likelihood ratio to take care ofthe discontinuity and using, say, the pathwise method, to evaluate the remainder. Weshall se examples of how to do this in the next section.5.5 Estimation in the Black-Scholes modelWe now consider the Black-Scholes model, given by the dynamicsdS t = µS t dt + σS t dW t ,endowed with a risk free asset B based on a constant short rate r with initial conditionB 0 = 1. We will prove that the model is free of arbitrage and implement the MonteCarlo methods discussed in the previous sections for the Black-Scholes model. Now,

5.5 Estimation in the Black-Scholes model 211our main goal of our numerical work is to analyze the performance of the Malliavinestimator and the Malliavin estimator combined with ordinary Monte Carlo methods,comparing it to other estimators. As we shall see, in the Black-Scholes case, theMalliavin method coincides with the Likelihood ratio method. Therefore, in this case,the Malliavin calculus brings nothing new at all. As a consequence, the Black-Scholescase is mostly a kind of warm-up case for us, allowing us to familiarize ourselveswith the various methods in a simple setting before turning to the somewhat morecumbersome case of the Heston model.Theorem 5.5.1. The Black-Scholes model is free of arbitrage, and there is a T -EMMyielding the dynamics dS t = rS t dt + σS t dW t .Proof. Both statements of the theorem follows immediately from Corollary 5.1.10.Comment 5.5.2 The T -EMM is in fact in this case unique, corresponding to completenessof the Black-Scholes model.◦For the numerical experiments, we consider three problems:1. The price of a call option.2. The price of a digital option.3. The delta of a digital option.It is easy to check for consistency of our methods, because in all three cases, the correctvalue has an analytical formula, as the following lemma shows.Lemma 5.5.3. Define d 1 (s) = 1σ √ T (log( s K ) + (r + 1 2 σ2 )T ) and d 2 (s) = d 1 (s) − σ √ T .1. The price of a strike K T -call option is sΦ(d 1 (S 0 )) − e −rT KΦ(d 2 (S 0 )).2. The delta of a strike K T -call option is Φ(d 1 (S 0 )).3. The delta of a strike K T -digital option is e −rT φ(d 2 (S 0 ))σ √ T S 0.Proof. The first result is the Black-Scholes formula, see Proposition 7.10 in Björk(2004). The second is Proposition 9.5 in Björk (2004). To prove the last result, we

212 Mathematical Financefirst calculate the price of a digital option ase −rT E Q (1 (ST ≥K))= e −rT Q(S T ≥ K)( ( 1= e −rT Qσ √ log S (TT s − ( 1= e(1 −rT − Φσ √ T= e −rT Φ(d 2 (S 0 )).From this we immediately obtainr − 1 ) )2 σ2 T(log K s − (r − 1 2 σ2 )T≥ 1σ √ T)))(log K (rs − − 1 ) ))2 σ2 TddS 0e −rT E Q (1 (ST ≥K)) = e −rT 1φ(d 2 (S 0 ))σ √ .T S 0We now proceed to the numerical experiments. All three problems all yield differingdegrees of irregularity, and we shall see that the methods applicable are different forall three cases.The price of a call option. Letting W be a Brownian motion and letting S bethe corresponding geometric Brownian motion with drift r and volatility σ, we wantto evaluate e −rT E(S T − K) + . We consider r = 4%, σ = 0.2, S 0 = 100, T = 1 andK = 104. We consider a plain Monte Carlo estimator, an antithetic estimator, astratified estimator, a control variate estimator and finally, an estimator combining allof these.The plain Monte Carlo estimator is simply implemented by simulating scaled lognormalvariables with scale S 0 , log mean (r − 1 2 σ2 )T and log variance σ 2 T . The antitheticestimator stratifies uniform variables and transforms them with the quantile functionof the lognormal distribution. The control variate estimator uses S T as a controlvariate, which has the known mean e rT S 0 and has a correlation with (S T − K) + ofaround 60%. To analyze the results, we know that all the estimators are unbiased, so itwill suffice to compare standard deviations. We base our standard deviation estimateson 80 Monte Carlo simulations for each method with batches from 2000 simulations to90000 simulations with step sizes of 2000. Since some methods use more computationaltime per simulation than others, we compare the standard deviations as functions ofcomputing time instead of as functions of number of simulations. The results canbe seen in Figure 5.3. We see that the stratified and the combined estimators havegraphs which contain no data before around 0.002 seconds of computing time. Thisis because we base our results on the same batches of simulations. Since these twomethods take significantly longer time per simulation than the other methods, their

5.5 Estimation in the Black-Scholes model 213Standard deviation0.00 0.05 0.10 0.15 0.20Plain MCAntitheticStratifiedControl variateCombined0.005 0.010 0.015 0.020Computation time (s)Figure 5.3: Comparison of standard deviations of Monte Carlo estimators for the priceof a call option in the Black-Scholes model.batches correspond to longer computational times. The effect is not particularly largehere, but the distinction between computing time and number of simulations willbecome important for example when considering the digital delta, where the mosttime-consuming method will spend six times as much computing time per simulationcompared to the simplest method. Comparing standard deviations as functions of thenumber of simulations would therefore have made the more advanced methods seemsix times more effective than they really are.The results show, consistently with the general rules of thumb set forth in Secion 4.7of Glasserman (2003), that stratified sampling yields the greatest benefit, followedby the control variate method and then antithetic sampling. Combining all of theseyields the best estimator. This estimator has a standard deviation which is up to 20times smaller than the ordinary Monte Carlo estimator.Importance sampling for OTM call options. So far, we have neglected theimportance sampling method. This is because for options which are not far out of themoney, it has little impact. For simplicity, we will in general only consider call anddigital options which are not far out of the money. For completeness, however, we nowillustrate how importance sampling can be used to facilitate computations for out ofthe money options.

214 Mathematical FinanceWe consider the same call as before, but change the strike from 104 to 225. Since theinitial value of the instrument is 100, this makes the call quite far out of the money.This means that many of the simulated values of S T will yield a payoff of zero, whichmakes the estimation of the mean difficult. We will use importance sampling to changethe distribution of the underlying normal distribution such that samples giving a payoffare more likely. To this end, we note that with d 2 (x) = 1σ √ T (log x S 0− (r − 1 2 σ2 )T ) andZ = W Tσ √ T , we have E(S T − K) + = E(d 2 (Zσ √ T ) − K) + . Let µ be the distribution ofZ, µ is then the standard normal distribution. Let ν be the normal distribution withmean ξ and unit variance. Let f µ and f ν be the corresponding densities. We thenhavedµdν (x) = f µ(x)f ν (x)exp ( − 1 2=x2)exp ( − 1 2(x − ξ)2)(= exp − 1 (x 2 − (x − ξ) 2))2= exp(−ξx + 1 )2 ξ2 .Letting Y be a normal distribution with mean ξ and unit variance, we therefore obtainE(S T − K) + = E(d 2 (Y σ √ T ) − K) + exp(−ξY + 1 )2 ξ2 ,which is the basis of the importance sampling estimator based on ν. By trial anderror experimentation, we find that ξ = 2 is a good choice for the problem at hand.Figure 5.4 compares estimated means and standard deviations for the ordinary MonteCarlo estimator, the importance sampling estimator and the importance samplingestimator with antithetic and stratified sampling and using Y as a control variate.Not surprisingly, we see that the importance sampling estimators are clearly superior.The combined estimator has a standard deviation which is up to 130 times smallerthan the ordinary estimator.The delta of a call option. Next, we consider the delta of a call option. Thissituation will allow us to use some of the Monte Carlo methods designed specificallyfor evaluation of sensitivities. We implement the finite difference method, the pathwisedifferentiation method and the likelihood ratio method. As we shall see, in this casethe pathwise differentiation method seems to be the optimal. We also consider anestimator combining the pathwise differentiation method with antithetic and stratifiedsampling. The methods for estimating sensitivities are not as straightforward to use

5.5 Estimation in the Black-Scholes model 215Standard deviation0e+00 2e−04 4e−04 6e−04 8e−04 1e−03Plain MCImportance SamplingCombined0.005 0.010 0.015 0.020Computation time (s)Figure 5.4: Comparison of means and standard deviations of Monte Carlo estimatorsfor an OTM call option in the Black-Scholes model.as those for plain Monte Carlo estimation. We will spend a short time reviewing theanalysis and implementation.As we saw in Section 5.4, to implement the finite difference method, we need to choosehow fast to let the denominator in the finite difference approximation tend to zero asthe number of simulations increase, and this choice depends on the variance of thenumerator of the finite difference apprroximation. In practice, we also need to choosethe constant factor c in h n = cn −γ . In our case, the numerator is V (f(S T (S 0 + h)) −f(S T (S 0 − h))), where S T (S 0 ) is the solution of the SDE for the financial instrumentwith initial value S 0 , and f is the call payoff. Our simulation method is to simulate ageometric Brownian motion with initial value one, multiply by S 0 + h and S 0 − h andtransform with the payoff function. f(S T (S 0 +h)) and f(S T (S 0 −h)) are then somewhatcorrelated. This both improves convergence and saves time generating simulations.When dicsussing the finite difference method in the last section, we gave argumentsfor different choices according to different situations. We begin by examining how muchof a difference these choices really makes. Figure 5.5 shows the results. Basically, thefigure shows that as long as our choices aren’t completely unreasonable, the differenceis quite small, in particular for large samples. The behaviour is consistent with a βvalue of 2. We pick γ = 1 6and c = 100. For the pathwise differentiation method, we

216 Mathematical FinanceStandard deviation0.000 0.005 0.010 0.015gamma = 1/4, initial = 0.01gamma = 1/5, initial = 0.01gamma = 1/6, initial = 0.01gamma = 1/4, initial = 1gamma = 1/5, initial = 1gamma = 1/6, initial = 1gamma = 1/4, initial = 100gamma = 1/5, initial = 100gamma = 1/6, initial = 1000.000 0.005 0.010 0.015 0.020SimulationsFigure 5.5: Comparison of standard deviations of Monte Carlo estimators for the deltaof a call option in the Black-Scholes model.wish to make the interchange∂E(S T − K) + = E ∂ E(S T − K) + .∂S 0 ∂S 0We need to check the conditions in Section 5.4. Since we have the explicit formulaS T = S 0 exp((r − 1 2 σ2 )T + σW T ), the variables for varying S 0 are clearly defined onthe same probability space, and S T is always differentiable as a function of S 0 . Sincex ↦→ (x − K) + is everywhere differentiable except in K and S T = K with probabilityzero for any value of S 0 , we conclude that (S T −K) + is almost surely differentiable withrespect to S 0 , and since ∂S T∂S 0= S TS0, a derivative is 1 [K,∞) (S T ) S TS0, which is integrable.To check that the interchange of differentiation and integration is allowed, we merelynote that with S T (S 0 ) = S 0 exp((r − 1 2 σ2 )T + σW T ),|(S T (S 0 + h) − K) + − (S T (S 0 ) − K) + | ≤ |h| exp((r − 1 ) )2 σ2 T + σW T ,using that x ↦→ (x − K) + is Lipschitz with constant 1. Since the coefficient to |h| is integrable,by the results of Section 5.4, the interchange of differentiation and integrationis allowed, and the corresponding pathwise estimator is based on E1 [K,∞) (S T ) S TS0.

5.5 Estimation in the Black-Scholes model 217For the likelihood ratio method, we want to make the interchange∂∂S 0E(S T − K) + = E(S T − K) +∂∂S 0g S0 (S T )g S0 (S T )to obtain the right-hand side as the basis for the likelihood ratio estimator, where g S0is the density of S T under the initial condition S 0 , which is given by the expressiong S0 (x) = 1xσ √ φ(d T2(S 0 , x)), with d 2 (S 0 , x) = 1σ √ (log x T S 0− (r − 1 2 σ2 )T ). We will nottry to justify the interchange, but merely conclude that numerical experiments showthat it in fact yields the correct result. To obtain a simpler expression, we note∂∂S 0g S0 (x)=g S0 (x)1xσ √ T1S 0 σ √ d T2(S 0 , x)φ(d 2 (S 0 , x))1xσ √ φ(d = d 2(S 0 , x)T2(S 0 , x))S 0 σ √ T ,which shows, using the representation of S T in terms of W T ,∂∂S 0g S0 (S T )= d 2(S 0 , S T )g S0 (S T ) S 0 σ √ = W TT S 0 σ √ T ,so that we need to calculate the mean E(S T − K) +W TS 0 σ √ . T,We mentioned in theintroduction to this section that the likelihood method and the Malliavin weightsmethod are equivalent for the Black-Scholes model. This is quite clear, since theMalliavin weights method yields∂E(S T − K) + = E(S T − K) + 1 ∫ T∂S 0 S 0 T 0which is precisely the same as the likelihood ratio method.1σ dW t = E(S T − K) + W TS 0 σ √ T ,We know that the pathwise differentiation and likelihood ratio methods are unbiased,but the finite difference method has a bias. Our simulation results show that this biasgenerally is between 10 −4 and 10 −5 . This is small enough that we can disregard it,and to compare the estimators, it will then suffice to compare standard deviations.We compare a finite difference estimator, a pathwise estimator, a likelihood ratioestimator and finally, a pathwise estimator with antithetic and stratified sampling. Asin the previous case, we base our standard deviation estimates on 80 Monte Carlosimulations for each method with batches from 2000 simulations to 90000 simulationswith step sizes of 2000. The results can be seen in Figure 5.6. We see that the finitedifference and the pathwise differentiation methods yield almost similar results. Whencombined with antithetic and stratified sampling, a considerable boost in effectivityis obtained. The likelihood ratio is in this case clearly inferior to the other methods,approximately doubling the standard deviation. This conclusion supports the common

218 Mathematical FinanceStandard deviation0.000 0.005 0.010 0.015Finite DifferencePathwise differentiationLikelihood ratioCombined0.005 0.010 0.015 0.020Computation time (s)Figure 5.6: Comparison of standard deviations of Monte Carlo estimators for the deltaof a call option in the Black-Scholes model.wisdom that the likelihood ratio method mostly is useful when applied in the contextof discontinuities. This conclusion is consistent with the results of Benhamou (2001)and Benhamou (2003). To see what goes wrong with the likelihood ratio method,recall that the likelihood ratio estimator is based on the formuladE(S T − K) + = E(S T − K) + W TdS 0 S 0 σ √ T .While the weight in the expectation on the right-hand side enables us to remove thederivative, it also introduces extra variance: The variable (S T − K) + W T gets verylarge when W T is large. We could use the localisation methods described in Section5.4 to minimize this effect, but in reality, the likelihood ratio method simply is notsuited to this particular problem. When considering the delta of a digital option, wewill see a situation where the likelihood ratio and localised likelihood ratio methodsprovide effective results.The delta of a digital option. Our final numerical experiment for the Black-Scholesmodel is the evaluation of the delta for a digital option. The methods which are apriori at our disposal are that of finite difference, pathwise differentiation and likelihoodratio. However, in this case, the pathwise differentiation method cannot be applied:Even though 1 [K,∞) (S T ) is almost surely differentiable for any initial condition, thederivative is also almost surely zero, so the pathwise differentiation method yields adelta for the digital option of zero, which is clearly nonsense. We conclude that the

5.5 Estimation in the Black-Scholes model 219interchange of differentiation and integration required for application of the pathwisedifferentiation method is not justified. Intuitively, this is because the payoff is notLipschitz. We are therefore left with the finite difference method and the likelihoodratio method. We implement these, and we also implement a localised version of thelikelihood ratio method. Finally, we also consider a localised likelihood ratio estimatorwith antithetic and stratified sampling.The finite difference and likelihood ratio methods are implemented as for the calldelta. We will discuss the details of the localisation of the likelihood method, wherewe proceed as described in Section 5.4. Let f(x) = 1 [K,∞) (x) be the digital payoff.Our goal is to make a decomposition of the form f(x) = h ε (x) + (f − h ε )(x), wheref − h ε is reasonably smooth and h ε has small support and contains the discontinuityof the payoff. Led by Glasserman (2003), Section 7.3, we define h ε by the relationf(x) − h ε (x) = min{1 − 1 }2ε max{0, x − K + ε}We have in this way ensured that f − h ε is continuous. We then obtainh ε (x) = 1 [K,∞) (x)(K − ε − x) − + 1 (−∞,K) (K + ε − x) + .h ε is a function with small support containing the discontinuity of the payoff. f − h εis continuous. See Figure 5.7 for illustrations of the decomposition of the payoff.Payoff−1.0 −0.5 0.0 0.5 1.0Payoff−1.0 −0.5 0.0 0.5 1.0Payoff−1.0 −0.5 0.0 0.5 1.080 90 100 110 120Underlying80 90 100 110 120Underlying80 90 100 110 120UnderlyingFigure 5.7: From left to right: The payoff of a digital option, the smoothed payofff − h ε and the jump part h ε .The likelihood ratio method applied to the h ε term then yields∂∂S 0E(S T − K) + =∂∂S 0Eh ε (S T ) +W T= Eh ε (S T )S 0 σ √ T +∂ E(f − h ε )(S T )∂S 0 ∂ E(f − h ε )(S T ).∂S 0

220 Mathematical FinanceWe will use pathwise differentiation on the second term. f − h ε is almost surelydifferentiable for any initial condition with∂∂S 0(f − h ε )(S T ) = 1 2ε 1 (|S T −K|

5.5 Estimation in the Black-Scholes model 221MethodCall priceImprovement factorPlain MC 1.0Antithetic 1.2-1.8Control variates 1.7-2.2Stratified 7-12Combined 14-22OTM Call pricePlain MC 1.0Importance sampling 20-37Combined 69-132Call deltaPlain finite difference 1.0Pathwise 0.8-1.5Likelihood ratio 0.3-0.5Combined 9-14Digital deltaPlain finite difference 1.0Likelihood ratio 5-10Localised likelihood ratio 16-29Combined 122-205Table 5.1: Comparison of results for estimation of the call price, call delta and digitaldelta in the Black-Scholes model.Conclusions. We have now analyzed various Monte Carlo methods for the price ofa call, the delta of a call and the delta of a digital in the context of the Black-Scholesestimator. The overarching conclusion of our efforts is that when applied properly,using more than just plain Monte Carlo methods for pricing and plain finite differencefor sensitivities can have a very large payoff.Before proceeding to our numerical experiments for the Heston model, we comparethe efficiency improvements for the various methods. Table 5.5 shows the factor ofimprovement for the standard deviations for the various methods implemented comparedto the simplest methods. The factors are calculated as the minimal and maximalfactors of improvement for the standard deviation for all computational times.

222 Mathematical FinanceWe see that in all cases, the addition of antithetic and stratified sampling adds considerablyto the cost effectiveness of the estimators in terms of standard deviation percomputing time. In particular, the combination of antithetic and stratified samplingwith the localised likelihood ratio method for the digital delta yields almost unbelievableresults. Our results show that this method allows one to calculate the digitaldelta with far more precision using the combined method with 2000 simulations thanusing the plain finite difference method with 90000 simulations.5.6 Estimation in the Heston ModelThe Heston model is a model for a single financial asset given bydS t = µS(t) dt + √ σ(t)S(t) √ 1 − ρ 2 dW 1 (t) + √ σ(t)S(t)ρ dW 2 tdσ t = κ(θ − σ t ) dt + ν √ σ t dW 2 (t),where (W 1 , W 2 ) is a two-dimensional Brownian motion, µ, κ, θ and ν are positiveparameters and ρ ∈ (−1, 1). We call this equation the Heston SDE for S and σ.We also assume given a risk-free asset B based on a constant short rate r. As inthe previous section, our mission is to argue that the model is free of arbitrage andanalyze different estimation methods in this model. In contrast to the previous section,it is not obvious that there exists a solution (S, σ) to the pair of SDEs describing themodel. The question of arbitrage is also considerably more difficult than in the previoussection. In order not to get carried away by too much theory, we will without proofapply some results from the theory of SDEs to obtain the results necessary for ourpurposes.Theorem 5.6.1. Let κ, θ and ν be positive numbers. Assume that 2κθ ≥ ν 2 . TheSDE given by dσ t = κ(θ − σ t ) dt + ν √ σ t+ dW t has a weak solution, unique in law,and the solution is strictly positive. In particular, the solution also satisfies the SDEdσ t = κ(θ − σ t ) dt + ν √ σ t dW t .Comment 5.6.2 The process solving the SDE is known as the Cox-Ingersoll-Rossprocess, from Cox, Ingersoll & Ross (1985). The criterion 2κθ ≥ ν 2 ensures that themean reversion level θ is sufficiently large in comparison to the volatility so that thevariability of the solution cannot drive the process below zero.◦Proof. The existence of a positive weak solution is proved in Example 11.8 of Jacobsen

5.6 Estimation in the Heston Model 223(1989). To prove uniqueness in law, we use the Yamada-Watanabe uniqueness theorem,Theorem V.40.1 of Rogers & Williams (2000b). This result states that the SDE isunique in law if the drift cofficient is Lipschitz and the volatility coefficient satisfies(σ(x) − σ(y)) 2 ≤ ρ(|x − y|) for some ρ : [0, ∞) → [0, ∞) with ∫ ∞ 10 ρ(x)dx infinite. Inour case, the drift is µ(x) = κ(θ − x), which is clearly Lipschitz as it has constantderivative. For the volatility, we have σ(x) = √ x + . We put ρ(x) = x and note thatfor 0 ≤ x ≤ y,(σ(x) − σ(y)) 2 = x − 2 √ x √ y + y ≤ y − x = ρ(|x − y|).For x, y ≤ 0, we clearly also have (σ(x) − σ(y)) 2 = 0 ≤ ρ(|x − y|). Finally, in the casex ≤ 0 ≤ y we find (σ(x) − σ(y)) 2 = y ≤ ρ(|x − y|). We conclude that for any x, y ∈ R,(σ(x) − σ(y)) 2 ≤ ρ(|x − y|). Obviously, ρ satisfies the integrability criterion. Thus, thehypotheses of the Yamada-Watanabe theorem are satisfied, and we have uniquenessin law.With Theorem 5.6.1 in hand, we can conclude that the volatility process of the Hestonmodel exists and is positive if 2κθ ≥ ν 2 . By positivity, we can also take its square root.Therefore, the SDE for the financial instrument in the Heston SDE is well-defined, andby Lemma 3.7.1, there exists a process S solving it. Because of these considerations,we shall in the following always assume 2κθ ≥ ν 2 .Next, we consider the question of arbitrage. Because the volatility is stochastic, wecannot directly apply the criterions of Section 5.1. With a bit of work, however, wecan use the results to obtain the desired conclusion.Theorem 5.6.3. The Heston model is free of arbitrage, and for any T > 0 there is aT -EMM Q yielding the dynamics on [0, T ] given bydS t = rS(t) dt + √ σ(t)S(t) √ 1 − ρ 2 dW 1 (t) + √ σ(t)S(t)ρ dW 2 tdσ t = κ(θ − σ t ) dt + ν √ σ t dW 2 (t),that is, the dynamics for the volatility is preserved, and the drift for the instrument ischanged from µ to r.Proof. We wish to use Corollary 5.1.8 to prove freedom from arbitrage. We thereforeneed to analyze the equationµ − r = √ σ t√1 − ρ2 λ 1 (t) + √ σ t ρλ 2 (t).

224 Mathematical FinanceTo solve this, we put λ 2 = 0. The equation reduces to µ − r = σ t√1 − ρ2 λ(t), wherewe for notational simplicity have removed the subscript from the lambda. Since σ ispositive, this equation has a unique solution given byλ t =µ − r√1 − ρ2 √ σ t.Since σ is a standard process, it is clear that λ is progressive. By continuity, λ is locallybounded and therefore in L 2 (W ). By Corollary 5.1.8, if we can prove that E(M) isa martingale with M t = − ∫ t0 λ t dW 1 t , the market is free of arbitrage. It is then clearthat the Q-dynamics of S and σ are as stated in the theorem.To this end, it will by Lemma 3.8.6 suffice to show that EE(M) t = 1 for all t ≥ 0. Lett ≥ 0 be given, let τ n = inf{s ≥ 0||λ s | ≥ n}. Since λ is continuous, τ n increases toinfinity and λ τ nis bounded. Then [M τn∧t ] ∞ = ∫ t0 (λτ n s ) 2 ds is bounded, so the Novikovcriterion applies to show that E(M τn∧t ) is a uniformly integrable martingale for anyt ≥ 0.Next, note that since τ n increases, τ n ≥ t from a point onwards and E(M) is nonnegative,the variables E(M) t 1 (τn ≥t) converge upwards to E(M) t . By monotone convergence,EE(M) t = lim n EE(M) t 1 (τn≥t). But E(M) t 1 (τn≥t) = E(M τn∧t ) ∞ 1 (τn≥t), soletting Q n be the measure such that dQn ∞dP ∞= E(M τn∧t ) ∞ , we may now concludeEE(M) t = limnEE(M) t 1 (τn ≥t) = limnEE(M τ n∧t ) ∞ 1 (τn ≥t) = limnQ n (τ n ≥ t).We need to show that the latter limit is equal to one. To do so, we first show thatQ n (τ n ≥ t) = P (τ n ≥ t). Under Q n , we know from Girsanov’s theorem that theprocess (Wt 1 + ∫ t0 λ s1 [0,τn ∧t](s) ds, Wt 2 ) is a F t Brownian motion. In particular, W 2 isa F t Brownian motion under Q n for all n. Now, we know that under P , σ satisfiesσ t = σ 0 +∫ t0κ(θ − σ s ) ds +∫ t0ν √ σ s dW 2 s .By Theorem 3.8.4, integrals under P and Q n are the same. Since ν √ σ s is continuous,it is integrable both under P and Q n , and therefore σ also satisfies the above equationunder the set-up with the Q n F t Brownian motion W 2 . Since the SDE satisfiesuniqueness in law by Theorem 5.6.1, the Q n distribution of σ must be the same asthe P distribution. Since λ is a transformation of σ and τ n is a transformation of λ,we conclude that the distribution of τ n is the same under Q n and P . In particular,Q n (τ n ≥ t) = P (τ n ≥ t). Since τ n tends almost surely to infinity, lim n P (τ n ≥ t) = 1.Combining this with our earlier findings, we conclude EE(M) t = 1. Therefore, byLemma 3.8.6, E(M) is a martingale and by Corollary 5.1.8, the market is free ofarbitrage.

5.6 Estimation in the Heston Model 225We will now discuss estimating the digital delta in the Heston model. Fix a maturitytime T . Our first objective is to find out how to simulate values from S T . In the Black-Scholes model, this was a trivial question as the stock price then followed a scaledlognormal distribution. For the Heston model, there is no simple way to simulate fromS T , we need to discretise the SDE. Before considering how to do this, we will rewritethe SDE to a form better suited for discretisation. Define X t = log S t , Itô’s lemma inthe form of Lemma 3.6.4 yieldsdX t = 1 S tdS t − 12S 2 td[S] t= r dt + √ σ t√1 − ρ2 dW 1 t + √ σ t ρ dW 2 t − 1 2 σ t dt,with X 0 = log S 0 . In other words, in order to simulate values of S T , it will suffice toconsider the SDE(dX t = r − 1 )2 σ t dt + √ √σ t 1 − ρ2 dWt 1 + √ σ t ρ dWt2dσ t = κ(θ − σ t ) dt + ν √ σ t dW 2 (t),find a way to simulate from X T and then take the exponent of X T to obtain a simulationfrom S T . A discretisation of the above set of SDEs will yield a way to simulatefrom a distribution approximating that of X T . Such a discretisation consists of definingdiscrete processes X n∆ ′ and σ′ n∆ for n ≤ N with N∆ = T such that the distributionof X ′ (T ) approximates that of X(T ). There are several ways to do this. Andersen(2008) provides a useful overview. We will consider two methods: The full truncationmodified Euler scheme and a scheme with exact simulation of the volatility. Forbrevity, we will denote the full truncation modified Euler scheme as the FT scheme,short for full truncation, and we will call the other scheme the SE scheme, short forsemi-exact.The FT scheme. We first describe the modified Euler scheme. In Lord et al. (2006),several ways of discretising the CIR process are considered, and the FT scheme isfound to be the one introducing the least bias. Let Z 1 and Z 2 be discrete processeson {0, ∆, . . . , N∆} of independent standard normal variables, the FT scheme is thediscretisation defined by putting X ′ (0) = X 0 , σ ′ (0) = σ 0 andX ′ (n+1)∆ = X′ n∆ +(r − 1 2 σ′ n∆σ ′ (n+1)∆ = σ′ n∆ + κ(θ − σ ′+n∆ ) + ν √σ ′+n∆√∆Z2n∆ .)√∆ + σn∆√ √ √′+ 1 − ρ2∆Zn∆ 1 + σ ′+n∆ ρ√ ∆Zn∆1The discretisation means that the volatility process σ ′ can turn negative. In this case,

226 Mathematical Financethere is an upwards drift of rate κθ. Taking positive parts ensure that the square rootsare well-defined.The SE scheme. The SE scheme takes advantage of the fact that the transitionprobability of the CIR process is a scaled noncentral χ 2 distribution. More explicitly,the distribution of σ t+∆ given σ ∆ = v is a scaled noncentral χ 2 distribution with scaleν 24κ (1 − e−κ∆ 4vκe), non-centrality parameter−κ∆4θκand degrees of freedomν 2 (1−e −κ∆ ) ν. That2this is the case is a folklore result, although it is quite unclear where to find a proofof the result. In Feller (1951), Lemma 9, the Fokker-Planck equation for the SDEis solved, yielding the density of the noncentral χ 2 distribution. It would then bereasonable to expect that one might obtain a complete proof by using some results onwhen the solution to the Fokker-Planck equation in fact is the transition density of thesolution. In any case, the result allows exact simulation of values of the CIR process.Thus, we can define σ ′ by letting the conditional distribution of σ(n+1)∆ ′ given σ′ n∆ bethe noncentral χ 2 distribution defined above. This defines a discrete approximation ofσ. We will use this to obtain a discrete approximation of the price process. To do so,we first note thatX t+∆ − X t = r∆ − 1 2∫ t+∆tσ u du + √ ∫ t+∆√1 − ρ 2 σu dWu 1 + ρt∫ t+∆t√σu dW 2 u.The process W 2 is correlated with σ. Since we desire to simulate directly from thetransition probabilities for σ, our discretisation scheme cannot depend on this correlation.To remove the dW 2 integral, we note thatσ t+∆ − σ t =∫ t+∆tκ(θ − σ u ) du + νIsolating the last integral and substituting, we find∫ t+∆t√σu dW 2 u.X t+∆ − X t= r∆ −∫ t+∆= r∆ − ρκθ∆νtσ u2 + ρκν (θ − σ u) dt + √ 1 − ρ 2 ∫ t+∆+ ρ(σ t+∆ − σ t )ν+( ρκν − 1 ) ∫ t+∆2tt√σu dW 1 u + ρ(σ t+∆ − σ t )νσ u du + √ ∫ t+∆√1 − ρ 2 σu dWu.1The right-hand side depends only on the joint distributions of σ u and W 1 . Sinceσ u and W 1 are independent, this is an important simplification compared to ourearlier expression, which included an integral with respect to W 2 . Because of theindependence, the conditional distribution of ∫ t+∆ √σu dW 1 tu given σ is normal witht

5.6 Estimation in the Heston Model 227mean zero and variance ∫ t+∆σt u du. We now make the approximationst∫ t+∆t∫ t+∆∆σ u du ≈ σ t+∆2 + σ ∆t2√√σu dWu1 ∆≈ σ t+∆2 + σ tInserting these in our equation for X t+∆ , we obtain∆21√ (Wt+∆ 1 − Wt 1 ).∆X t+∆ ≈ X t + r∆ − ρκθ∆ + ρ(σ t+∆ − σ t )ν ν( ρκ+ν − 1 )∆(σt+∆ + σ t )+ √ √σt+∆1 − ρ2 22 + σ t(Wt+∆ 1 − Wt 1 )2(= X t + r − ρκθ ) ( ( ∆ ρκ∆ +ν 2 ν − 1 )− ρ )σ t2 ν( ( ∆ ρκ+2 ν − 1 )+ ρ ) √(1 − ρ2 )(σ t+∆ + σ t )σ t+∆ +(Wt+∆ 1 − Wt 1 ).2 ν2We are thus lead to defining the discretisation scheme X ′ for X byX ′ (n+1)∆ = X′ n∆ + K 0 + K 1 σ ′ n∆ + K 2 σ ′ (n+1)∆ + √K 3 (σ ′ (n+1)∆ + σ′ n∆ )Z n,where Z n is a process of independent standard normal variables on {0, ∆, . . . , N∆}independent of σ ′ and(K 0 = r − ρκθ )∆νK 1 = ∆ ( ρκ2 ν − 1 )− ρ 2 νK 2 = ∆ ( ρκ2 ν − 1 )+ ρ 2 νK 3 = ∆(1 − ρ2 ).2Modulo a short rate and some weights in the discretisation of the integrals, this is thesame as obtained in Andersen (2008), page 20. To sum up, the SE scheme defines theapproximation σ ′ to σ by directly defining the conditional distributions of the processusing the transition probabilities. The approximation X ′ to X is then obtained byX ′ (n+1)∆ = X′ n∆ + K 0 + K 1 σ ′ n∆ + K 2 σ ′ (n+1)∆ + √K 3 (σ ′ (n+1)∆ + σ′ n∆ )Z n,where Z n is a process of independent standard normal variables on {0, ∆, . . . , N∆},independent of σ ′ . Note that while the FT scheme simulates a discretisation of thetwo-dimensional process (W 1 , W 2 ), the SE scheme simulates only a discretisation of

228 Mathematical FinanceW 1 . This will be an important point when comparing our Malliavin estimator withthe one obtained in Benhamou (2001). Also note that because of the exact simulationof the root volatility in the SE scheme, σ ′ can never reach zero, but in the FT scheme,it is quite probable that σ ′ hits zero. This will be important for the implementationof the Malliavin estimator.Performance of the schemes. We will now see how well each of these two schemescan approximate the true distribution of S T . As we have no way to obtain exact simulationsfrom the distribution, we must contend ourselves with checking convergenceand compare results from each scheme. We will use S 0 = 100, σ 0 = 0.04, r = 0.04,κ = 2, θ = 0.04, ν = 0.4, ρ = 0.3 and T = 1 and these parameter values will beheld constant throughout all of this section unless stated otherwise. Figure 5.9 showsdensity estimates of S T for each of the two schemes, using 2, 5, 50 and 100 steps. Thefigure shows that both schemes seem to converge to some very similar distributions.Most of the error in the discretisations seem to come from the top shape of the hump,with the SE scheme converging quicker than the FT scheme. This is not surprising,considering that the SE scheme uses the exact distribution for the volatility, whilethe FT scheme only has an approximation. Both the schemes seem to have achievedstability at least at the 50-step discretisation.Density0.000 0.005 0.010 0.015 0.020 0.0252−step FT scheme5−step FT scheme50−step FT scheme100−step FT schemeDensity0.000 0.005 0.010 0.015 0.020 0.0252−step SE scheme5−step SE scheme50−step SE scheme100−step SE scheme0 50 100 150 200 2500 50 100 150 200 250ValueValueFigure 5.9: Density estimates for the FT and SE schemes based on 80000 simulations.Next, we compare call pricing under each of the schemes. In Heston (1993), an analyticalexpression for the Fourier transform of the survival function of S T is obtained,and it is explained how this can be inverted and used to obtain call prices in the Heston

5.6 Estimation in the Heston Model 229model. These results greatly contribute to the popularity of the Heston model andits offspring. We use the pricer at http://kluge.in-chemnitz.de/tools/pricer to obtainaccurate values for call prices. This pricer seems to be reliable and is also used for thenumerical experiments in Alòs & Ewald (2008). Using strike 104 yields a call price of7.71175.We want to see how the call prices obtained using Monte Carlo with each of thetwo schemes compare to this. The purpose of this is both to check how much bias isintroduced by the discretisation and to check that our fundamental simulation routineswork as they are supposed to. We will at the same time also check the functionalityof sampling from each of the schemes using latin hypercube sampling and antitheticsampling. We have explained earlier how antithetic sampling works, but we havenot mentioned latin hypercube sampling. We will not explain this method in detail,suffice it to say that it is can be thought of as a generalization of stratified samplingfor high-dimensional problems, see Glasserman (2003), Section 4.4.In the left graph of Figure 5.10, we compare the relative errors of call prices for eachof the two schemes with and without latin hypercube and antithetic sampling, andwe compare the standard deviations. We use discretisations steps from 10 to 100and consider 300 replications of the estimator, each estimate based on 20000 samples.We see that as expected, each of the methods yield prices which fit well with thetrue value. It seems that less than 50 steps yields some inaccuracy in excess of theexpected variability, with the 10-step results giving bias which is significantly largerthan those of the observations for higher levels of discretisation. However, the relativeerror never exceeds 1%. In reality, models are inaccurate anyway, so this bias would beunimportant in practical settings. In other words, for practical applications, it wouldseem that 20 steps would by sufficient for the contract under investigation.In the right graph, we compare standard deviation times computation time. Thisis a general measure of efficiency, with low values corresponding to high efficiency.We see that the efficiency decreases as the number of discretisation steps increases.This is not particularly surprising, but an important point nonetheless: The higherlevel of discretisation means that each sample from S T takes longer to generate, butsince the number of samples is held constant and the change in the distribution isminuscule, the standard deviation does not decrease. Thus, the standard deviationtimes computational time will increase as the number of discretisation steps increases,lowering performance. Therefore, the choice of number of discretisation steps boilsdown to choosing between optimising bias and optimising standard deviation times

230 Mathematical Financecomputational time.Note that the right graph cannot be used to measure the efficiency of latin hypercubeand antithetic sampling, as the computational time is not held constant. However,experimental results show an improvement factor of around 1.4. This is very moderatecompared to the improvement factors found for antithetic and stratified sampling inthe previous section. The reason for this is probably that the transformation fromthe driving normal distributions to the final S T sample is very complex, and the latinhypercube sampling basically just stratifies in a very simple manner. To obtain thesame type of efficiency gains for the Heston model as in the Black-Scholes model, it willprobably be necessary to implement a more sophisticated form of stratified sampling,such as terminal stratification or one of the other methods discussed in Subsection4.3.2 of Glasserman (2003).Relative error−0.010 −0.005 0.000 0.005 0.010FT scheme with plain Monte CarloFT scheme with Latin Hypercube and AntitheticsSE scheme with plain Monte CarloSE scheme with Latin Hypercube and AntitheticsStandard deviation times computational time (s)0.0 0.1 0.2 0.3 0.4 0.5FT scheme with plain Monte CarloFT scheme with Latin Hypercube and AntitheticsSE scheme with plain Monte CarloSE scheme with Latin Hypercube and Antithetics20 40 60 80 10020 40 60 80 100Discretisation stepsDiscretisation stepsFigure 5.10: Left: Relative errors for call prices calculated using the FT and SEschemes with and without latin hypercube sampling and antithetic sampling. Right:Standard deviations times computational time.Comparison of methods for calculating the digital delta. Being done withthe introductory investigations of the Heston model, we now proceed to the mainpoint of this section, which is the comparison of the Malliavin and localised Malliavinmethods with the finite difference method. Note that we cannot directly comparethese methods with the likelihood ratio and pathwise differentiation methods as inthe previous section, as the likelihood ratio method does not apply directly to theHeston model (see, however, the arguments outlined in Glasserman (2003), page 414or Chen & Glasserman (2006), for application of the likelihood ratio method to thediscretisation of the Heston model) and the pathwise differentiation method does not

5.6 Estimation in the Heston Model 231apply to the digital delta. In the remainder of this section, we will always sample fromour two schemes using latin hypercube sampling and antithetic sampling.We consider a digital option with strike 104. Let f denote the corresponding payofffunction. As described above, we have six methods to consider: The finite differencemethod, the Malliavin method and the localised Mallivin method, each combined withone of the two simulation schemes. In Section 5.4, we found the basis of the Malliavinmethod through the identity∂∂S 0Ef(S T ) = E(f(S T ) 1S 0 T∫ T0)1√ √ dW 1σt 1 − ρ2t ,where we are using √ σ t instead of σ t because in our model formulation, σ t is theroot volatility and not the ordinary volatility. The localised Malliavin method is thenobtained by decomposing the payoff in two parts and applying the Malliavin methodto one part and the pathwise differentiation method to the other, as was also donein Section 5.5. One problem with this is that we do not know how to sample fromthe stochastic integral in the expectation above. Instead, we have to make to with anapproximation. Based on the theory of Chapter 3, we would expect that a reasonableapproximation of an integral of the form ∫ T0 H s dW 1 s is∫ T0H s dW 1 s =n∑H tk−1 (W tk − W tk−1 ),k=1where it is essential that we are using forward increments and not backwards incrementsin the Riemann sum. With X ′ and σ ′ denoting the discretised versions of Xand σ and Z 1 denoting the driving Brownian motion for the asset price process, it isthen natural to consider the approximation⎛∂Ef(S T ) ≈ E ⎝f∂S 0( )e X′ T1e X′ 0T √ 1 − ρ 2N∑k=1√∆Zk∆√σ ′ (k−1)∆This works well when using the SE scheme, where we know that σ ′ is positive. However,if we are using the FT scheme, we have no such guarantee, and the above is not welldefined.In this case, then, we resort to substituting max{σ ′ (k−1)∆ , ε} for σ′ (k−1)∆ inthe sum, where ε is some small positive number. Our experimental results show thatusing an ε which is too small such as 0.00000001 can lead to considerable numericalinstability. We use 0.0001, this leads to stable results and reasonable efficiency.⎞⎠ .We are now ready for the numerical experiments. We begin by investigating the biasof our estimators. We expect two sources of bias. First, there is an inherent bias

232 Mathematical Financefrom the sampling of S T since we are only sampling from a distribution approximatingthat of S T . Second, each of the methods involve certain approximations - the finitedifference method approximates the derivative by a finite difference, and the Malliavinestimators approximate the Itô integral by a sum.In Figure 5.11, we have plotted the means for each of the six methods. We use 300replications of estimators based on samples of size 1000 to 29500 with increments of1500. We repeat the experiment for a 15-step and a 100-step discretisation. We seethat in the 15-step experiment, there is a considerable variation in the means, but thisvariation is almost gone in the 100-step experiment. Expecting that the finite differencemethod produces the least bias, we obtain from the experiment an approximatevalue for the digital delta of 0.02132436, we will use this value as a benchmark. Notethat we could also have used the semi-analytical results from Heston (1993) to obtainbenchmark values, however, the results from the finite difference method will besufficient for our needs. Using this benchmark, we find that the Malliavin methods inthe 15-step experiment yields average relative errors of -6.8% and 7.7% percent for theFT and SE schemes, respectively. The localised methods have average relative errorsof -3.3% and 3.7%, while the finite difference relative errors are less than 0.1%. In the100-step experiment, the Malliavin methods have average relative errors of -0.8% and1.9% for the FT and SE schemes, respectively, while the localised Malliavin methodshave average relative errors of -0.3% and 0.9%, respectively. We conclude that it wouldseem that all of our estimators are asymptotically unbiased.Mean0.010 0.015 0.020 0.025 0.030SE Finite differenceFT Finite differenceSE MalliavinFT MalliavinLocalised SE MalliavinLocalised FT MalliavinMean0.010 0.015 0.020 0.025 0.030SE Finite differenceFT Finite differenceSE MalliavinFT MalliavinLocalised SE MalliavinLocalised FT Malliavin0.0 0.5 1.0 1.50 1 2 3 4 5 6 7Computation time (s)Computation time (s)Figure 5.11: Estimated estimator means for estimating the digital delta. 15-stepdiscretisation on the left, 100-step discretisation on the right.

5.6 Estimation in the Heston Model 233Next, we consider standard deviation. The results can be found in Figure 5.12. Forboth the 15-step and 100-step experiments and both the FT and SE schemes, we seethat the Malliavin method is superior to the finite difference method and the localisedMalliavin method is superior to the ordinary Malliavin method.For the 15-step experiment, the SE scheme yields the better performance, while thesituation is reversed for the 100-step experiment. The explanation for this probablyhas something to do with the FT scheme being faster than the SE scheme since itis faster to generate normal distributions than noncentral χ 2 distributions. Thus, forfine discretisations, the resulting distributions of S T are nearly the same, but the FTscheme is computationally faster.Standard deviation0.000 0.001 0.002 0.003 0.004 0.005 0.006SE Finite differenceFT Finite differenceSE MalliavinFT MalliavinLocalised SE MalliavinLocalised FT MalliavinStandard deviation0.000 0.001 0.002 0.003 0.004 0.005 0.006SE Finite differenceFT Finite differenceSE MalliavinFT MalliavinLocalised SE MalliavinLocalised FT Malliavin0.0 0.5 1.0 1.50 1 2 3 4 5 6 7Computation time (s)Computation time (s)Figure 5.12: Estimated estimator standard deviations for estimating the digital delta.15-step discretisation on the left, 100-step discretisation on the right.The difference between our results and those of Benhamou (2001). Wementioned in Section 5.4 that the expression we have obtained for the Malliavin methodis not the same as is obtained in Benhamou (2001). We will now explain what exactlythe difference is and investigate how this difference manifests itself in practice.We will first detail the model description and corresponding Malliavin estimator usedby ourselves and by Benhamou (2001). Our description of the Heston model and the

234 Mathematical Financecorresponding Malliavin estimator isdS t = µS(t) dt + √ σ(t)S(t) √ 1 − ρ 2 dW 1 (t) + √ σ(t)S(t)ρ dW 2 tdσ t = κ(θ − σ t ) dt + ν √ σ t dW 2 (t)(f(S T ) 1 ∫ TS 0 T∂∂S 0Ef(S T ) = E01√σt√1 − ρ2 dW 1 twhere (W 1 , W 2 ) is a two-dimensional Brownian motion. In contrast to this, Benhamou(2001) list the model description and Malliavin estimator as)dS t = µS(t) dt + σ(t)S(t) dB 1 (t)dσt 2 = κ(θ − σt 2 ) dt + νσ t dB 2 (t)(∂Ef(S T ) = E f(S T ) 1 ∫ )T1dBt1 ,∂S 0 S 0 T σ twhere (B 1 , B 2 ) is a two-dimensional Brownian motion with correlation ρ. Rewritingthis with our model formulation, we obtain0,dS t = µS(t) dt + √ σ(t)S(t) √ 1 − ρ 2 dW 1 (t) + √ σ(t)S(t)ρ dW 2 tdσ t = κ(θ − σ t ) dt + ν √ σ t dW 2 (t)( (∂Ef(S T ) = E f(S T ) 1 √1− ρ2∂S 0 S 0 T∫ T0∫1T√ dWt 1 + ρσt 0))1√ dWt2 .σtObviously, there seems to be some difference here. The two expressions for the Malliavinestimator are equal in the case of zero correlation, but different whenever thereis nonzero correlation. We will now see how this difference manifests itself when calculatingthe digital delta. We will compare the results of the estimator of Benhamou(2001) with our estimators and the finite difference estimator. Since the estimator ofBenhamou (2001) involves an integral with respect to both W 1 and W 2 , we cannotuse the SE scheme with this method. We therefore restrict ourselves to comparing theMalliavin estimator of Benhamou (2001) under the FT scheme with our Malliavinestimator under both the FT and SE schemes. Finally, we also consider the finitedifference method under the SE scheme. Since we have already seen that the finitedifference method is quite insensitive to the choice of discretisation scheme, it shouldbe unnecessary to consider also the finite difference method under the FT scheme.We compare the digital delta estimators when the correlation varies between -0.95 and0.95. We use 300 replications of estimators based on 20000 simulations each. Theresults can be seen in the left graph of Figure 5.13. As expected, the results from

5.6 Estimation in the Heston Model 235Benhamou (2001) fits with our results in the case of zero correlation. However, it isquite obvious that when the correlation is nonzero, in particular when there is verystrong positive or negative correlation, the estimator from Benhamou (2001) does notfit with the otherwise very reliable finite difference estimator, whereas our Malliavinestimator works well.To do a more formal check that there is an actual difference, we have in the rightgraph of Figure 5.13 plotted density estimates for the finite difference estimator andthe Benhamou estimator for the case of correlation −0.95. The thin lines are momentmatchednormal densities. We see that the estimators are approximately normal, withthe finite difference estimator having some minor fluctuations in the right tail. Itwould therefore be reasonable to use the Welch t-test for testing equality of means fornormal distributions with unequal variances to test whether the two estimators havedifferent means. The p-value of the test comes out less than 10 −15 . For comparison,when using the same test to compare our FT scheme based Malliavin estimator withthe finite difference estimator, we obtain a p-value of 71%. This strongly suggests thateven when taking the bias from the approximation to the Itô integral in the Benhamouestimator into account, the Benhamou estimator does not yield the correct value ofthe digital delta.Mean0.016 0.018 0.020 0.022 0.024LA Finite DifferenceFT MalliavinLA MalliavinFT BenhamouDensity0 100 200 300 400 500Finite differenceBenhamou−1.0 −0.5 0.0 0.5 1.00.016 0.018 0.020 0.022 0.024CorrelationValueFigure 5.13: Left: Comparison of the estimator from Benhamou (2001) with ourMalliavin estimators and a finite difference estimator. Right: Density plots for thefinite difference and Benhamou estimators for correlation -0.95. Thick lines are densityestimates, thin lines are normal approximations.

236 Mathematical FinanceMethod15-step digital deltaImprovement factorFT scheme, finite difference 1.0SE scheme, finite difference 0.9-1.1FT scheme, Malliavin 1.0-1.4SE scheme, Malliavin 1.0-2.5FT scheme, localised Malliavin 1.6-2.4SE scheme, localised Malliavin 2.6-3.8100-step digital deltaFT scheme, finite difference 1.0SE scheme, finite difference 0.9-1.1FT scheme, Malliavin 2.3-3.1SE scheme, Malliavin 1.3-2.4FT scheme, localised Malliavin 3.4-5.1SE scheme, localised Malliavin 2.7-3.8Table 5.2: Comparison of results for estimation of the digital delta in the Hestonmodel.Conclusions. We have considered the digital delta for three types of estimators, eachcombined with two different discretisation schemes. As in the previous section, weconclude that methods more exotic than just plain Monte Carlo with finite differencecan make a considerable difference. Table 5.5 shows the factor of improvement for thestandard deviations for the various methods implemented compared to the simplestmethods.We see that the Malliavin and localised Malliavin methods indeed yield an improvement,but the improvement is somewhat disappointing compared to the results weobtained for the Black-Scholes model, where the likelihood ratio and localised likelihoodratio methods yielded improvement factors for the digital delta of respectively 5to 10 and 16 to 29. It is difficult to say why the improvement factors are so small inthis case. The Malliavin method requires no extra simulations, only the calculation ofthe approximation to the Itô integral, which is merely a simple sum.We also again note that while the SE scheme is most effective for the 15-step case,the FT scheme is most effective for the 100-step scheme. Now, the localised Malliavinmethod for the 15-step case yielded a bias of up to 3.7%, which may be a little much.One might get more satisfactory results using, say, a 30-step discretisation. In this

5.7 Notes 237case, the SE scheme is probably still the most effective. We conclude that for practicalpurposes, the SE scheme should be preferred above the FT scheme.5.7 NotesThe main references for the material in Section 5.1 are Björk (2004) and Øksendal(2005), Chapter 12. Other useful resources are Karatzas & Shreve (1998) and Steele(2000). Our results are not as elegant as they could be because of the limited theoryavailable to us. In general, the result of Theorem 5.1.5 on the existence of EMMsand freedom from arbitrage goes both ways, in the sense that under certain regularityconditions and appropriate concepts of freedom from arbitrage, a market is free fromarbitrage if and only if there exists an EMM. For a precise result, see for exampleDelbaen & Schachermayer (1997), Theorem 3. The theory associated with the necessityof the existence of EMMs in order to preclude arbitrage is rather difficult and ispresented in detail in Delbaen & Schachermayer (2008). The simplest introduction tothe field seems to be Levental & Skorohod (1995).All of the Monte Carlo methods introduced in Section 5.3 and Section 5.4 except theresults on the Malliavin method are described in the excellent Glasserman (2003).Rigorous resources on the Malliavin calculus applied to calculation of risk numbersare hard to come by. Even today, the paper Fournié et al. (1999) seems to be thebest, because of its very clear statement of results. Di Nunno et al. (2008) is alsorefreshingly explicit. Other papers on the applications of the Malliavin calculus forcalculations of risk numbers are Fournié et al. (2001), which extends the investigationsstarted in Fournié et al. (1999), and also Benhamou (2001) and Benhamou (2003). Avery interesting paper is Chen & Glasserman (2006), which proves, using results fromthe theory of weak convergence of SDEs, that the Malliavin method in a reasonablylarge class of cases is equivalent to the limit of an average of likehood ratio estimatorsfor Euler discretisations of the SDEs under consideration. Montero & Kohatsu-Higa(2002) provides an informal overview of the role of Malliavin calculus in the calculationof risk numbers. Nualart (2006), Chapter 6, also provides some resources.Other applications of the Malliavin calculus in finance such as the calculation of conditionalexpectations, insider trading and explicit hedging strategies are discussed inSchröter (2007), Di Nunno et al. (2008), Nualart (2006) and Malliavin & Thalmaier(2005). Here, Schröter (2007) is very informal, and Malliavin & Thalmaier (2005) is

238 Mathematical Financeextremely difficult to read.Much of the SDE theory used in the derivation of the Malliavin estimator in Section5.4 and Fournié et al. (1999) is described in Karatzas & Shreve (1988) and Rogers& Williams (2000b). Readable results on differentiation of flows is hard to find. Theclassical resource is Kunita (1990). A seemingly more readable proof of the mainresult can be found in Ikeda & Watanabe (1989), Proposition 2.2 of Chapter V.Several of the numerical results of Section 5.5 can be compared with those of Fourniéet al. (1999) and Benhamou (2001). Glasserman (2003) is, as usual, a useful generalguide to the effectivity of Monte Carlo methods.The proof in Section 5.6 that the Heston model does not allow arbitrage is based onthe techniques of Cheridito et al. (2005), specifically the proof of Theorem 1 in thatpaper. Numerical results on the performance of Malliavin methods for the calculationof risk numbers in the Heston model are, puzzlingly, very difficult to find, actuallyseemlingly impossible to find. Also, the only paper giving an explicit form of theMalliavin weight for the Heston model seems to be Benhamou (2001), so it is unclearwhether it is generally understood, as we concluded, that the form given is erroneous.There are, however, several other papers on applications of the Malliavin calculus tothe Heston model, but these applications are not on the calculation of risk numbers.See for example Alòs & Ewald (2008).Pertaining discretisation of the Heston model, Andersen (2008) seems to be the bestresource available, and contains references to many other papers on the subject. Foran curious paper giving a semi-analytical formula for the density of the log of the assetprice in the Heston model, see Dragulescu & Yakovenko (2002). This could perhapsbe useful as a benchmark for the true density.The methods for calculating risk numbers considered here all have one major advantageover the finite difference method, an advantage which is not seen in our context. In ourcase, we only consider risk numbers with respect to changes in the initial value S 0 of thethe underlying asset. Since S T is linear in S 0 , we can in the finite difference methodreuse our simulations for the estimation of both means by scaling our simulations.Therefore, in spite of the finite difference method requiring the calculation of twomeans instead of just one, as in the other methods we consider, the computational timespent on simulations is not much different from the other methods. For risk numberswith respect to general parameters, the connection between S T and the risk number

5.7 Notes 239variable is not as straightforward, and in such cases we would have to simulate twice asmany variables in the finite difference method as for the other methods for calculatingrisk numbers. When having to calculate many different risk numbers, which is oftenthe case in practice, this observation can have a major impact on computational time.All of the numerical work in this chapter is done in R. We also considered usingthe language OCaml instead, hoping for improvements in both elegance and speed.However, in the end, the mediocrity of the IDE available for OCaml and the strongand efficient statistical library available for R made R the better choice.

240 Mathematical Finance

Chapter 6ConclusionsIn this chapter, we will discuss the results which we have obtained in all three mainparts of the thesis, and we will consider opportunities for extending this work.6.1 Discussion of resultsWe wil discuss the results obtained for stochastic integrals, the Malliavin calculus andthe applications to mathematical finance.Stochastic integrals. In Chapter 3, we successfully managed to develop the theoryof stochastic integration for integrators of the form A t + ∑ n∫ tk=1 0 Y tk dWt k , where Ais a continuous process of finite variation and Y k ∈ L 2 (W ), and we proved the basicresults of the theory without reference to any external results. However, the increase inrigor compared to Øksendal (2005) or Steele (2000) came at some price, resulting insome honestly very tedious work. Furthermore, while we managed to prove Girsanov’stheorem without reference to external results, the resulting proof was rather longcompared to the modern proofs based on stochastic integration for general continuouslocal martingales.We must therefore conclude that the ideal presentation of the theory would be basedon general continuous local martingales. And with the proof of the existence of the

242 Conclusionsquadratic variation of Section C.1, the biggest difficulty of that theory could be convenientlyseparated from the actual development of the integral.The Malliavin calculus. We presented in Chapter 3 the basic results on the Malliavinderivative. While we only managed to cover a very small fraction of the resultsin Nualart (2006), we have managed to develop the proofs in much higher detail,including the development of some results used in the proofs such as Lemma 4.4.19which curiously are completely absent from other accounts of the theory. The valueof this work is considerable, clarifying the fundamentals of the theory and thereforepaving the way for higher levels of detail in further proofs as well.Also, the extension of the chain rule proved in Theorem 4.3.5 is an important newresult. Considering how fundamental the chain rule is to the theory, this extensionshould make the Malliavin derivative somewhat easier to work with.Mathematical finance. Chapter 5 contained our work on mathematical finance. Wemanaged to use the theory developed Chapter 3 to set up a basic class of financialmarket models and proved sufficient conditions for absence of arbitrage, demonstratingthat the theory of stochastic integration developed in Chapter 3 was sufficiently richto obtain reasonable results. The use of the Theorem 3.8.4 in the proof of Theorem5.1.5 and the Itô representation theorem in the proof of Theorem 5.1.11 were crucial,details which are often skipped in other accounts.We also made an in-depth investigation of different estimation methods for pricesand sensitivities in the Black-Scholes model, documenting the high effectivity of moreadvanced Monte Carlo methods and confirming results found in the literature. Weapplied the Malliavin methods to the Heston, obtaining new results. We saw that theefficiency gains seen in the Black-Scholes case could not be sustained in the Hestonmodel. We also noted a problem with the Malliavin estimator given Benhamou (2001)and corrected the error.6.2 Opportunities for further workAll three main parts of the thesis presents considerable oppportunities for further work.We will now detail some of these.

6.2 Opportunities for further work 243Stochastic integration for discontinuous FV processes. It is generally acceptedthat while the theory of stochastic integration for continuous semimartingales is somewhatmanageable, the corresponding theory for general semimartingales is very difficult,as can be seen for example in Chapter VI of Rogers & Williams (2000b) or inProtter (2005). Nonetheless, financial models with jumps are becoming increasinglypopular, see for example Cont & Tankov (2004). It would therefore be convenient tohave a theory of stochastic integration at hand which covers the necessary results. Tothis end, we observe that the vast majority of financial models with jumps in use onlyuses the addition of a compound poisson process as a driving term in, say, the assetprice. Such a process is of finite variation. Therefore, if it were possible to develop atheory of stochastic integration for integrators of the form A+M where A is a possiblydiscontinuous process of finite variation and M is a continuous local martingale, onecould potentially obtain a simplification of the general theory which would still be versatileenough to cover most financial applications. A theory alike to this is developedin Chapter 11 of Shreve (2004).Measurability results. There are several problems relating to measurability propertiesin the further theory of the Malliavin calculus. As outlined in Section 4.5, if uis in the domain of the Skorohod integral with u t ∈ D 1,2 for all t ≤ T and there aremeasurable versions of (t, s) ↦→ D t u s and ∫ T0 D tu s δW s , then∫ tD t u s δW s =0∫ T0D t u s δW s + u t .It would be useful for the theory if there were general criteria to ensure that all of themeasurability properties necessary for this to hold were satisfied. Another problemraises its head when considering the Itô-Wiener expansion. Also described in Section4.5, this is a series expansion for variables F ∈ L 2 (F T ), stating that there exists squareintegrabledeterministic functions f n on ∆ n such that we have the L 2 -summationformulaF =∞∑n=0∫ T ∫ tn∫ t3∫ t2n! · · · f n (t) dW (t 1 ) · · · dW (t n ).0 0 0 0Here, it is not trivial to realize that the iterated integrals are well-defined. To see thenature of the problem, consider defining the single iterated integral∫ T ∫ t200f(t 1 , t 2 ) dW (t 1 ) dW (t 2 ).Fix t 2 ≤ T . The process ∫ t0 f(t 1, t 2 ) dW (t 1 ) is well-defined if t 1 ↦→ f(t 1 , t 2 ) is progressivelymeasurable. However, for varying t 2 , the processes ∫ t0 f(t 1, t 2 ) dW (t 1 ) are

244 Conclusionsbased on different integrands, and there is therefore a priori no guarantee that thesecan be combined in such a way as to make t 2 ↦→ ∫ t 20 f(t 1, t 2 ) dW (t 1 ) progressivelymeasurable, which is necessary to be able to integrate this process with respect tothe Brownian motion. Thus, it is not clear at all how to make the statement of theItô-Wiener expansion rigorous. A further problem is that in the proof of the expansionbased on the Itô representation theorem, it is necessary to know that the integrandsin the representations can be chosen in certain measurable ways.The simplest problems are those involving the Wiener-Itô expansion. As usual, puttingθ(f) = ∫ T0 f(t) dW t, θ is a continuous mapping from L 2 [0, T ] to L 2 (F T ). Furthermore,if we let R(X) denote the Itô representation of a variable X ∈ L 2 (F T ) in the sensethatX =∫ T0R(X) t dW t ,then R is a continuous linear mapping from L 2 (F T ) to L 2 (Σ π [0, T ]). Thus, the problemsrelated to the Wiener-Itô expansion are both in some sense about iterating continuouslinear operators over L 2 spaces. In contrast to this, the problems related to theMalliavin derivative are in some sense about selecting measurable versions of closedlinear operators over subspaces of L 2 spaces.It is not clear how to solve these problems in general. However, a first step would be toshow a result which for example would allow us to, given a process X indexed by [0, T ] 2and being B[0, T ] 2 ⊗ Σ π [0, T ] measurable, select a version of (s, t) ↦→ ∫ tX(s, t) dW (t)0which is B[0, T ] 2 ⊗ Σ π [0, T ] measurable. Applying this several times in an appropriatemanner could help defining the iterated integrals, and it is also a result of this typenecessary in the Malliavin calculus.In order to cover all of our cases, let us consider a mapping A : L 2 (E, E) → L 2 (K, K)between two L 2 spaces. We endow these L 2 spaces with their Borel-σ-algebras B(E)and B(K) and assume that A is B(E)-B(K) measurable. This obviously covers thecases from the Wiener-Itô expansion, since continuous mappings are measurable. Andby Corollary, 4.4.16, the Malliavin derivative is continous on H n for each n ≥ 0,where L 2 (F T ) = ⊕ ∞ n=0H n , leading us to suspect that the Malliavin derivative indeedis measurable as well on its domain.Let (D, D) be another measurable space. We will outline how to prove the followingresult: If X : D × E → R is D ⊗ E measurable such that for each x ∈ D it holds thatX(x, ·) ∈ L 2 (E, E), there exists a D ⊗ K measurable version of (x, y) ↦→ A(X(x, ·))(y).

6.2 Opportunities for further work 245This can be directly applied to obtain measurable versions of stochastic integrandswhere only one coordinate is integrated.Let X : D × E → R be given with the properties stated above. We first defineY : D → L 2 (E, E) by putting Y (x)(y) = X(x, y). By a monotone class argument,we prove that Y is D-B(E) measurable. We can then use that A is measurable toobtain that x ↦→ A(Y (x)) is D-B(K) measurable. Defining Z(x, z) = A(Y (x))(z), Zis a mapping from D × K to R. By definition, Z(x, ·) = A(Y (x)) = A(X(x, ·)). Weare done if we can argue that there exists a measurable version of Z. This is the mostdifficult part of the argument. It is easy in the case where x ↦→ A(Y (x)) is simple.For the general case, we use that L 2 convergence implies almost sure convergence fora subsequence. Selecting a subsequence for each x in a measurable manner, we canobtain a measurable version. The argument is something like that found in Protter(2005), Theorem IV.63. However, the technique of that proof depends on the linearityof the integral mapping. We need to utilize measurability properties directly, makingthe proof more cumbersome.Risk numbers. In our investigation of the Malliavin method for the Heston model,we only considered the digital delta. Furthermore, we only applied a few extra MonteCarlo methods to our routines, namely latin hypercube sampling and antithetic sampling.There are several other methods available, such as those of Fries & Yoshi (2008)or those detailed in Glasserman (2003), which would be interesting to attempt to applyto the Heston model.Furthermore, it would be interesting to investigate the efficiency gains possible forpath-dependent options, both of european, bermudan or american type, and it wouldbe interesting to extend the results to other models. Here, the extension to modelswith jumps poses an extra challenge, as the Malliavin calculus we have developed onlyworks in the case of Brownian motion. For Malliavin calculus for processes with jumpssuch as Lévy processes, see Di Nunno et al. (2008).

246 Conclusions

Appendix AAnalysisIn this appendix, we develop the results from pure analysis that we will need. Wewill give some main theorems from analysis, and we will consider the basic resultson spaces L p (µ), Cc∞ (R n ) and C ∞ (R n ). We also consider Hermite polynomials andHilbert spaces. The appendix is more or less a smorgasbord, containing a large varietyof different results. For some results, references are given to proofs. For other results,full proofs are given. The reason for using so much energy on these proofs is basicallythat there does not seem to be any simple set of sources giving all the necessary results.Our main sources for the real analysis are Carothers (2000), Berg & Madsen (2001),Rudin (1987) and Cohn (1980). For Hilbert space theory, we use Rudin (1987),Hansen (2006) and Meise & Vogt (1997).A.1 Functions of finite variationWe say that a function F : [0, ∞) → R has finite variation ifn∑V F (t) = sup |F (t k ) − F (t k−1 )|k=1is finite for all t > 0, where the supremum is taken over all partitions of [0, t]. In thiscase, we call V F the variation of F over [0, t]. We will give the basic properties offunctions of finite variation. This will be used when defining the stochastic integral

248 Analysisfor integrators which have paths of finite variation.Lemma A.1.1. V F is continuous if and only of F is continuous.Proof. See Carothers (2000), Theorem 13.9.Theorem A.1.2. There is a unique measure µ F such that µ F (a, b] = F (b) − F (a) forany 0 ≤ a ≤ b < ∞.Proof. See Theorem 3.2.6 of Dudley (2002) for the case where F is monotone. Thegeneral case follows easily by applying the decomposition of F into a difference ofmonotone functions.Lemma A.1.3. If F is continuous, then µ F has no atoms.Proof. This follows directly from the definition of µ F given in Theorem A.1.2.Since functions of finite variation corresponds to measures by Theorem A.1.2, we candefine the integral with respect to a function of finite variation.Definition A.1.4. We put L 1 (F ) = L 1 (µ F ) and define, for any function x ∈ L 1 (F ),∫ t0 x u dF u = ∫ t0 x u dµ F (u).Lemma A.1.5. Let F be continuous and of finite variation. If g is continuous, theng is integrable with respect to F over [0, t], and the Riemann sumsn∑g(t k−1 )(F (t k ) − F (t k−1 ))k=1converge to ∫ tg(s) dF (s) as the mesh of the partition tends to zero.0Proof. We haven∑g(t k−1 )(F (t k ) − F (t k−1 )) =k=1n∑g(t k−1 )µ F ((t k−1 , t k ]) =k=1∫ t0s n dµ F ,where s n = ∑ nk=1 g(t k−1)1 (tk−1 ,t k ]. As the mesh tends to zero, s n tends to g by theuniform continuity of g on [0, t]. Since g is continous, it is bounded on [0, t]. Therefore,by dominated convergence, ∫ t0 s n dµ F tends to ∫ t0 g(s) dµ F (s), as desired.

A.2 Some classical results 249Lemma A.1.6. If x ∈ L 1 (F ), then t ↦→ ∫ t0 x u dF (u) has finite variation.Proof. This follows by decomposing F into a difference between monotone functionsand applying the linearity of the integral in the integrator.A.2 Some classical resultsIn this section, we collect a few assorted results. These will be used in basicallyevery part of the thesis. Gronwall’s lemma is used in the proof of Itô’s representationtheorem, the multidimensional mean value theorem is used in various approximationarguments, the Lebesgue differentiation theorem, is used in density arguments for theItô integral and the duality result on L p and L q is used when extending the Malliavinderivative.Lemma A.2.1 (Gronwall). Let g, b : [0, ∞) → [0, ∞) be two measurable mappingsand let a be a nonnegative constant. Let T > 0 and assume ∫ Tb(s) ds < ∞. If0g(t) ≤ a + ∫ t0 g(s)b(s) ds for t ≤ T , then g(t) ≤ a exp(∫ tb(s) ds) for t ≤ T .0Proof. Define B(t) = ∫ t0 b(s) ds and G(t) = ∫ tg(s)b(s) ds. Then0ddt e−B(t) G(t) = e −B(t) b(t)g(t)−e −B(t) G(t)b(t) = e −B(t) b(t)(g(t)−G(t)) ≤ ae −B(t) b(t)for t ≤ T . Integrating this equation from 0 to T , we finde −B(t) G(t) ≤∫ tThis shows G(t) ≤ a(e B(t) − 1) and thereforeas was to be proved.g(t) ≤ a +0∫ t0(ae −B(s) b(s) ds = a 1 − e −B(t)) .g(s)b(s) ds = a + G(t) ≤ ae B(t) ,Lemma A.2.2. Let ϕ ∈ C 1 (R). For any distinct x, y ∈ R n , there exists ξ on the linebetween x and y such thatϕ(y) − ϕ(x) =n∑k=1∂ϕ∂x k(ξ k )(y k − x k ).

250 AnalysisProof. Define g : [0, 1] → R n by g(t) = ty + (1 − t)x. Then, by the ordinary meanvalue theorem and the chain rule, for some θ ∈ [0, 1],as desired.f(y) − f(x) = (f ◦ g)(1) − (f ◦ g)(0)= (f ◦ g) ′ (θ)n∑ ∂f= (g(θ))g∂xk(θ)′k=k=1n∑k=1∂f∂x k(θy + (1 − θ)x)(y k − x k ),Theorem A.2.3 (Lebesgue’s differentiation theorem). Let f ∈ L 1 (R) and putF (x) = ∫ xf(y) dy. Then F is almost everywhere differentiable with derivative f.−∞Proof. See Rudin (1987), Theorem 7.11 and Theorem 7.7.Corollary A.2.4. Let f : R → R be bounded. Then it holds thatalmost everywhere for t ∈ R.lim 1 1n∫ tt− 1 nf(x) dx = f(t)Proof. Consider any bounded interval (a, b), and put g (a,b) = f(x)1 (a,b) (x). Theng ∈ L 1 (R), and theorem A.2.3 yields for almost all t ∈ (a, b) thatlim 1 1n∫ tt− 1 nf(x) dx = lim 1 1n∫ tt− 1 ng(x) dx = g(t) = f(t).Since a countable union of null sets is again a null set, the conclusion of the lemmafollows.Theorem A.2.5. Let p, q ≥ 1 be dual exponents and let (E, E, µ) be a measure space.The dual of L p (E, E, µ) is isometrically isomorphic to L q (E, E, µ), and an isometricisomorphism ϕ : L q (E, E, µ) → L p (E, E, µ) ′ is given by∫ϕ(G)(F ) = G(x)F (x) dµ(x).In particular, an element F ∈ L p (E, E, µ) is almost surely zero if and only if it holdsthat ∫ G(x)F (x) dµ(x) = 0 for all G ∈ L q (E, E, µ).

A.3 Dirac families and Urysohn’s lemma 251Proof. For the duality statement and the isometric isomorphism, see Proposition 13.13of Meise & Vogt (1997). The last statement of the theorem follows from Proposition6.10 of Meise & Vogt (1997).A.3 Dirac families and Urysohn’s lemmaIn this section, we wil develop a version of Urysohn’s Lemma for smooth mappings.We first state the original Urysohn’s Lemma.Theorem A.3.1 (Urysohn’s Lemma). Let K be a compact set in R n , and let V bean open set. Assume K ⊆ V . There exists f ∈ C c (R n ) such that K ≺ f ≺ V .Proof. This is proven in Rudin (1987), Theorem 2.12.The mappings which exist by Theorem A.3.1 are called bump functions. Our goalis to extend Theorem A.3.1 to show the existence of bump functions which are notonly continuous, but are smooth. To do so, recall that for two Lebesgue integrablemappings f and g, the convolution f ∗ g is defined by (f ∗ g)(x) = ∫ f(x − y)g(y) dy.Fundamental results on the convolution operation can be found in Chapter 8 of Rudin(1987). The convolution of f with g has an averaging effect. The following two resultswill allow us to use this averaging effect to smooth out nonsmooth functions, allowingus to prove the smooth version of Urysohn’s lemma.Lemma A.3.2. If f : R n → R is Lebesgue integrable and g ∈ C ∞ (R n ), then theconvolution f ∗ g is in C ∞ (R n ).Proof. By definition, (f ∗ g)(x) = ∫ f(y)g(x − y) dy. Now, since g is differentiable,g is bounded on bounded sets and therefore, by Theorem 8.14 in Hansen (2004b),(f ∗g)(x) is differentiable in the k’th direction with ∂∂x k(f ∗g)(x) = ∫ f(y) ∂g∂x k(x−y) dy.Here ∂g∂x kis in C ∞ (R n ), so it follows inductively that f ∗ g ∈ C ∞ (R n ).Theorem A.3.3 (Dirac family). Let µ be a Radon measure on R n and let ‖ · ‖ beany norm on R n . There exists a family of mappings ψ ε ∈ C ∞ (R n ) such that ψ ε ≥ 0,suppψ ε ⊆ B ε (0) and ∫ ψ ε (x) dµ(x) = 1. Here, B ε (0) is the ε-ball in ‖ · ‖. We call (ψ ε )a Dirac family with respect to ‖ · ‖ and µ.

252 AnalysisProof. Since all norms on R n are equivalent, it will suffice to prove the result for ‖ · ‖ 2 .Therefore, B ε (0) will in the following denoted the centered ε-ball in ‖ · ‖ 2 . As seenin Berg & Madsen (2001), page 190, there exists χ ∈ C ∞ (R n ) such that 0 ≤ χ ≤ 1and suppχ ⊆ B 1 (0). Then χ ε given by χ ε (x) = χ( x ε ) is also in C∞ (R n ), 0 ≤ χ ε ≤ 1and suppχ ε ⊆ B ε (0). Since µ is Radon, any open ball in R n has finite µ-measure, andtherefore χ ε ∈ L 1 (µ). Putting ψ ε = 1‖χ ε ‖ 1χ ε , ψ ε then has the desired properties.Theorem A.3.4 (Smooth Urysohn’s Lemma). Let K be a compact set in R n , andlet V be an open set. Assume K ⊆ V . There exists f ∈ C ∞ c (R n ) such that K ≺ f ≺ V .Proof. By Theorem A.3.1, there exists f ∈ C c (R n ) such that K ≺ f ≺ V . Letε ∈ (0, 1 2) and choose δ parrying ε for the uniform continuity of f. Define the setsK ′ = f −1 ([1 − ε, 1]) and V ′ = f −1 (ε, ∞).Since K ≺ f ≺ V , clearly K ⊆ K ′ ⊆ V ′ ⊆ V . Furthermore, K is compact and V isopen. If x is such that ‖y − x‖ ≤ δ for some y ∈ K, then |f(x) − 1| = |f(x) − f(y)| ≤ εand therefore f(x) ≥ 1 − ε, so x ∈ K ′ . On the other hand, if x ∈ V ′ then f(x) > ε,so if ‖y − x‖ ≤ δ, |f(y) − f(x)| ≤ ε and therefore f(y) > 0, showing y ∈ V . In otherwords, B δ (x) ⊆ V .Now let g ∈ C c (R n ) with K ′ ≺ g ≺ V ′ and let h = g ∗ψ δ . We claim that h satisfies theproperties in the theorem. By Lemma A.3.2, h ∈ C ∞ c (R n ). We will show that h ≺ V .Let x ∈ V , we need to prove that h(x) = 0. We have h(x) = ∫ g(y)ψ δ (x − y) dy. Now,whenever ‖x − y‖ ≤ δ, the integrand is zero because ψ δ (x − y) is zero. On the otherhand, if ‖x − y‖ > δ, we must have y /∈ V ′ , because if y ∈ V ′ , B δ (y) ⊆ V . Therefore,g(y) is zero and we again find that the integrand is zero. Thus, h(x) is zero and h ≺ V .Next, we show that K ≺ h. Let x ∈ K. Whenever ‖y − x‖ ≤ δ, we have y ∈ K ′ andtherefore g(y) is one. On the other hand, whenever ‖y − x‖ > δ, ψ δ is zero. Thus,h(x) = ∫ g(y)ψ δ (x − y) dy = 1 and K ≺ h. We conclude K ≺ h ≺ V , as desired.We are also going to need a version of Urysohn’s lemma for closed sets instead ofcompact sets.Lemma A.3.5. Let F be a closed set in R n , and let V be an open set. Assume F ⊆ V .There exists f ∈ C ∞ (R n ) such that F ≺ f ≺ V .Proof. Defining f(x) =d(x,V c )d(x,F )+d(x,V c ), we find that f is continuous and F ≺ f ≺ V .

A.4 Approximation and density results 253Using the same technique as in the proof of Theorem A.3.4, we extend this to providesmoothness of the bump function.A.4 Approximation and density resultsIn this section, we will review an assortment of approximation and density resultswhich will be used throughout the thesis. In particular the Malliavin calculus makesgood use of various approximation results.Whenever our approximation results involve smooth functions, we will almost alwaysapply a Dirac family as a mollifier and take convolutions. The next three results areapplications of this technique, showing how to approximate respectively continuous,Lipschitz and a class of differentiable mappings.Lemma A.4.1. Let f ∈ C(R n ), and let ψ ε be a Dirac family. Then f ∗ ψ ε convergesuniformly to f on compacts.Proof. Let M > 0. For any x with ‖x‖ 2 ≤ M, we have|f(x) − (f ∗ ψ ε )(x)| =≤∫∫∣ f(x)ψ ε (x − y) dy − f(y)ψ ε (x − y) dy∣∫|f(x) − f(y)|ψ ε (x − y) dy≤sup |f(x) − f(y)|.y∈B ε (x)We may therefore concludesup |f(x) − (f ∗ ψ ε )(x)| ≤‖x‖ 2 ≤Msupsup‖x‖ 2 ≤M y∈B ε (x)|f(x) − f(y)|.Since f is continuous, f is uniformly continuous on compact sets. In particular, f isuniformly continuous on B M+δ (0) for any δ > 0, and therefore the above tends tozero.Lemma A.4.2. Let f : R n → R be a Lipschitz mapping with respect to ‖ · ‖ ∞ suchthat |f(x) − f(y)| ≤ K‖x − y‖ ∞ . There exists a sequence of mappings (g ε ) in C ∞ (R n )with partial derivatives bounded by K such that g ε converges uniformly to f.

254 AnalysisProof. Let (ψ ε ) be a Dirac family on R n with respect to ‖ · ‖ ∞ and define g ε = f ∗ ψ ε .Then g ε ∈ C ∞ (R n ), and we clearly have, with e k the k’th unit vector,|g ε (x + he k ) − g ε (x)| =≤≤∫∫∣ f(x + he k − y)ψ ε (y) dy − f(x − y)ψ ε (y) dy∣∫|f(x + he k − y) − f(x)|ψ ε (y) dyKh.It follows that g ε has partial derivatives bounded by K. It remains to show uniformconvergence of g ε to f, but this follows since|f(x) − g ε (x)| =≤≤≤∫∫∣ f(x)ψ ε (x − y) dy − f(y)ψ ε (x − y) dy∣∫|f(x) − f(y)|ψ ε (x − y) dy∫K ‖x − y‖ ∞ ψ ε (x − y) dyεK.The proof is finished.Lemma A.4.3. Let f ∈ C 1 (R n ), and assume that f has bounded partial derivatives.The exists a sequence (g ε ) ⊆ C ∞ p (R n ) with the following properties:1. g ε converges uniformly to f.2. ∂ xk g ε converges pointwise to ∂ xk f for k ≤ n.3. ‖∂ xk g‖ ∞ ≤ ‖∂ xk f‖ ∞ for k ≤ n.Proof. Let (ψ ε ) be a Dirac family on R n . Define g ε = f ∗ ψ ε . We then have g ε ∈C ∞ (R n ). We claim that g ε satisfies the properties of the lemma.

A.4 Approximation and density results 255Step 1: Uniform convergence. We find∫∫|f(x) − g ε (x)| =∣ f(x)ψ ε (x − y) dy − f(y)ψ ε (x − y) dy∣∫≤ |f(x) − f(y)|ψ ε (x − y) dy∫ ∣ n ∣∣∣∑ ∂f≤(ξ k )(x k − y k )∂x k ∣ ψ ε(x − y) dyk=1∫≤ max∂fk≤n∥ (ξ k )∂x k∥ ‖x − y‖ 1 ψ ε (x − y) dy∞≤√ ∫n max∂fk≤n∥ (ξ k )∂x k∥ ‖x − y‖ 2 ψ ε (x − y) dy∞≤ ε √ n max∂fk≤n∥ (ξ k )∂x k∥ ,∞so g ε converges uniformly to f.Step 2: Pointwise convergence of partial derivatives. We have∂g ε(x) =∂ ∫∫∂f(x − y)ψ ε (y) dy = f(x − y)ψ ε (y) dy = (∂ xk f) ∗ ψ ε .∂x k ∂x k ∂x kBy the same calculations as in the first step, we therefore find∫|∂ xk g ε (x) − ∂ xk f(x)| ≤ |∂ xk f(x) − ∂ xk f(y)|ψ ε (x − y) dy≤sup |∂ xk f(x) − ∂ xk f(y)|,y∈B ε (x)where B ε (x) denotes the ε-ball around x in ‖·‖ 2 . Continuity of ∂ xk f shows the desiredpointwise convergence.Step 3: Boundedness of partial derivatives. To show the boundedness conditionon the partial derivatives of g ε , we calculate∣ ∫∣∂g ε(x)∂x k∣ =∂∣ f(x − y)ψ ε (y) dy∂x k∣∫ ∣ ∣∣∣ ∂f≤ (x − y)∂x k∣ ψ ε(y) dy∥ ≤∂f ∥∥∥∞∥ ,∂x kshowing the final claim of the lemma. This also shows g ε ∈ C ∞ p (R n ).

256 AnalysisWe now prove that for any Radon measure µ on R n and any p ≥ 1, the space C ∞ c (R n )is dense in L p (µ).Lemma A.4.4. Cc∞ (R n ) is uniformly dense in C c (R n ). Furthermore, for any elementf ∈ C c (R n ), the approximating sequence in Cc∞ (R n ) can be taken to have a commoncompact support.Proof. Let f ∈ C c (R n ), and let ψ ε be a Dirac family on R n . Let M > 0 be suchthat suppf ⊆ B M (0). Put f ε = f ∗ ψ ε . By Lemma A.3.2, f ε ∈ C ∞ (R n ). And sincesuppψ ε ⊆ B ε (0), it holds that for x with ‖x‖ 2 ≥ M + ε,∫∫f ε (x) = f(y)ψ ε (x − y) dy = 1 BM (0)(y)1 Bε (x)(y)(x − y) dy = 0,so f ε has compact support. Therefore f ε ∈ Cc∞ (R n ). By Lemma A.4.1, f ε convergesuniformly to f on compacts. Since suppf ε ⊆ B M+ε (0), our approximating sequencehas a common compact support. In particular, the uniform convergence on compactsis actually true uniform convergence. This shows the claims of the lemma.Theorem A.4.5. Let µ be a Radon measure on R n . Then C ∞ c (R n ) is dense in L p (µ)for p ≥ 1.Proof. First note that by combining Theorem 2.14, Theorem 2.18 and Theorem 3.14of Rudin (1987), we can conclude that C c (R n ) is dense in L p (µ). It will thereforesuffice to show that we can approximate elements of C c (R n ) in L p (µ) with elementsof Cc ∞ (R n ). Let f ∈ C c (R n ). By Lemma A.4.4, there exists f n ∈ Cc ∞ (R n ) converginguniformly to f n , and by the same lemma, we can assume that all the functions f n andf have the same compact support K. Then,∫lim |f(x) − f n (x)| p dµ(x) ≤ lim ‖f − f n ‖ ∞ µ(K) = 0.nnA.5 Hermite polynomialsIn this section, we present and prove some basic results on Hermite polynomials. Asource for results such as these is Szegö (1939), Chapter V, where a good deal ofresults on Hermite polynomials are gathered. Most results are presented without

A.5 Hermite polynomials 257proof, however. Also note that what is called Hermite polynomials in Szegö (1939) isa little bit different than what is called Hermite polynomials here.Definition A.5.1. The n’th Hermite polynomial is defined byWe put H 0 (x) = 1 and H −1 (x) = 0.H n (x) = (−1) n e x22d n x2e− 2dxn .It is not completely obvious from Definition A.5.1 that the Hermite polynomials arepolynomials at all. Before showing that this is actually the case, we will obtain arecursion relation for the Hermite polynomials.Lemma A.5.2. It holds thatexp( )tx − t2 =2∞∑n=0H n (x)t n ,n!so that for any x, H n (x) is the n’th coefficient of the Taylor expansion of exp(tx − t2 2 )as a function of t. In particular,)∣H n (x) =(tx ∂n∂t n exp − t2 ∣∣∣t=0.2Proof. The mapping t ↦→ exp(− 1 2 (x − t)2 ) is obviously entire when considered as acomplex mapping, and therefore given by its Taylor expansion. We then obtain( ) (exp tx − t2 x2= exp22 − 1 )2 (x − t)2∞∑= e x2 t n d n2(−n! dt n exp 1 )∣∣∣∣t=02 (x − t)2= e x22==∞∑n=0∞∑n=0n=0∞∑ t n dn(−1)n(−n! dt n exp 1 )∣∣∣∣t=x2 t2 n=0t n(n! e x22 (−1)n dndx n exp − 1 )2 x2H n (x)t n .n!Lemma A.5.3. It holds that for n ≥ 0, H ′ n(x) = nH n−1 (x).

258 AnalysisProof. We see that∞∑n=0H n(x)′ t n = dn! dx∞∑n=0H n (x)t nn!)= d (txdx exp − t2 2( )= t exp tx − t2 2==∞∑n=0∞∑n=1H n (x)t n+1n!nH n−1 (x)t n .n!Recalling that H −1 (x) = 0, uniqueness of coefficients yields the result.Lemma A.5.4. It holds that for n ≥ 0, H n+1 (x) = xH n (x) − nH n−1 (x).Proof. We find, using the product rule and Lemma A.5.3,H n+1 (x) = (−1) n+1 e x22as was to be proved.= (−1) n+1 e x22= (−1) n+1 ( ddxd n+1 x2e− 2dxn+1 d d n x2e− 2ndx(dxe x22= −H ′ n(x) + (−1) n xe x22= −nH n−1 (x) + xH n (x),d n x2e− 2dxn d n x2e− 2dxn ) ( ) )d−dx e x2 dnx22 e− 2dxn Lemma A.5.2, Lemma A.5.3 and Lemma A.5.4 along with the definition are the basicworkhorses for producing results about the Hermite polynomials. In particular, wecan now show that the Hermite polynomials actually are polynomials.Lemma A.5.5. The n’th Hermite polynomial H n is a polynomial of degree n.Proof. We use complete induction. The result is clearly true for n = 0. Assume thatit is true for all k ≤ n. We know that H n+1 (x) = xH n (x) − nH n−1 (x), by LemmaA.5.4. Here, nH n−1 (x) is a polynomial of degree n − 1 by assumption, and H n is a

A.6 Hilbert Spaces 259polynomial of degree n. Therefore, xH n (x) is a polynomial of degree n + 1, and theresult follows.Lemma A.5.6. The span of the n first Hermite polynomials H 0 , . . . , H n−1 is the sameas the span of the n first monomials 1, x, x 2 , . . . , x n−1 .Proof. From Lemma A.5.5, it is clear that the span of the n first monomials includesthe n first Hermite polynomials. To show the reverse inclusion, we use induction.Since H 0 (x) = 1, the result is trivially true for n = 1. Assume that it has been provenfor n. Let H n+1 be the span of the first n + 1 Hermite polynomials. We need to provethat x k ∈ H n+1 for k ≤ n. By our induction hypothesis, x k ∈ H n+1 for k ≤ n − 1. Toprove that x n ∈ H n+1 , note that by Lemma A.5.5, we know that H n is a polynomialof degree n, that is,n∑H n (x) = a k x k ,where a n ≠ 0. Since x k ∈ H n+1 for k ≤ n − 1, we obtain a n x n ∈ H n+1 . Since a n ≠ 0,this implies x n ∈ H n+1 , as desired.k=0A.6 Hilbert SpacesIn this section, we review some results from the theory of Hilbert spaces. Our mainsources are Hansen (2006) and Meise & Vogt (1997).Definition A.6.1. A Hilbert Space H is a complete normed linear space over R suchthat the norm of H is induced by an inner product.In the remainer of the section, H will denote a Hilbert space with inner product 〈·, ·〉and norm ‖ · ‖. We say that two elements x, y ∈ H are orthogonal and write x⊥y if〈x, y〉 = 0. We say that two subsets M and N of H are orthogonal and write M⊥Nif it holds that for each x ∈ M and y ∈ N, x⊥y. The elements orthogonal to M aredenoted M ⊥ . We write span M for the linear span of M, and we write span M forthe closure of the span of M.Lemma A.6.2. Let M and N be subsets of H. If M and N are orthogonal, then soare span M and span N.

260 AnalysisProof. By the linearity of the inner product, span M and span N are orthogonal. Bythe continuity of the inner product, span M and span N are orthogonal as well.Lemma A.6.3. If U is a subspace of H, then U ⊥ is a closed subspace of H.Proof. See Rudin (1987), p. 79.Lemma A.6.4. Let M is a subspace of H. M is dense in H if and only if M ⊥ = {0}.Proof. This follows from Corollary 11.8 of Meise & Vogt (1997).Theorem A.6.5 (Pythagoras’ Theorem). Let x 1 , . . . , x n be orthogonal elementsof H. Then∥ n∑ ∥∥∥∥2 n∑x∥ k = ‖x k ‖ 2 .k=1k=1Proof. See Hansen (2006), Theorem 3.1.14.Theorem A.6.6. Let U be a closed subspace of H. Each element x ∈ H has a uniquedecomposition x = P x + x − P x, where P x ∈ U and x − P x ∈ U ⊥ . The mappingP : H → U is a linear contraction and is called the orthogonal projection onto U.Proof. The existence of the decomposition and the linearity of P is proved in Rudin(1987), Theorem 4.11. That P is a contraction follows by applying Theorem A.6.5 toobtain‖P x‖ 2 ≤ ‖P x‖ 2 + ‖x − P x‖ 2 = ‖P x + x − P x‖ 2 = ‖x‖ 2 .Lemma A.6.7. Let U be a closed subspace of H. The orthogonal projection P ontoU satisfies P 2 = P , R(P ) = U and N(P ) = U ⊥ . Also, ‖x − P x‖ = d(x, U), where ddenotes the metric induced by ‖ · ‖.Proof. Let x ∈ H. Then P x ∈ U, and therefore P 2 x = P x, showing the first claim.Since P maps into U and P x = x for x ∈ U, it is clear that range satisfies R(P ) = U.To show the claim about the null space, is is clear that N ⊥ ⊆ N(P ). To show theother inclusion, let x ∈ N(P ). Then x = P x + x − P x = x − P x ∈ U ⊥ , as desired.

A.6 Hilbert Spaces 261Finally, the formula ‖x − P x‖ = d(x, U) follows by Lemma 11.5 of Meise & Vogt(1997).If M and N are orthogonal subspaces of H, we define the orthogonal sum M ⊕ N byM ⊕ N = {x + y|x ∈ M, y ∈ N}.Lemma A.6.8. If M and N are closed orthogonal subspaces of H, then M ⊕ N is aclosed subspace of H as well. Letting P and Q be the orthogonal projections onto Mand N, the orthogonal projection onto M ⊕ N is P + Q.Proof. That M ⊕ N is closed is proved in Lemma 3.5.8 of Hansen (2006). Let anelement x ∈ H be given. We write x = (P + Q)x + x − (P + Q)x, and if we can arguethat x − (P + Q)x ∈ (M ⊕ N) ⊥ , we are done, since (P + Q)x obviously is in M ⊕ N.To this end, first note that we have x = P x + (x − P x), where x − P x ∈ M ⊥ bythe definition of the orthogonal projection P . Using the definition of the orthogonalprojection Q, we also have x − P x = Q(x − P x) + (x − P x − Q(x − P x)), wherex − P x − Q(x − P x) ∈ N ⊥ . Since R(Q) = N, Q(x − P x) ∈ M ⊥ . We concludethat x − P x − Q(x − P x) is orthogonal to both N and M. In particular, then, it isorthogonal to M ⊕ N.Finally, by Lemma A.6.7, N(Q) = N ⊥ ⊇ M = R(P ). Thus, QP = 0 and we obtainx − (P + Q)x = x − P x − Q(x − P x) ∈ (M ⊕ N) ⊥ ,as desired.Lemma A.6.9. Letting M and N be orthogonal subspaces of H, it holds that M ⊕ Nis the span of M ∪ N.Proof. Clearly, M ⊕ N ⊆ span M ∪ N. On the other hand, if x ∈ span M ∪ N, wehave x = ∑ nk=1 λ kx k where x k ∈ M ∪ N. Splitting the sum into parts in M and partsin N, we obtain the reverse inclusion.Lemma A.6.10. Let x n be an orthogonal sequence in H. ∑ ∞n=1 x n is convergent ifand only if ∑ ∞n=1 ‖x n‖ 2 is finite.

262 AnalysisProof. Both directions of the equivalence follows from the relation, with n > m,∥ n∑ m∑ ∥∥∥∥2 n∑x∥ k − x k = ‖x k ‖ 2k=1k=1combined with the completeness of H.k=m+1We are also going to need infinite sums of orthogonal subspaces. If (U n ) is a sequenceof orthogonal subspaces of H, we define the orthogonal sum ⊕ ∞ n=1U n of the subspacesby{∞⊕∞∣ }∑ ∣∣∣∣ ∞∑U n = x n x n ∈ U n and ‖x n ‖ 2 < ∞n=1n=1This is well-defined, since we know from Lemma A.6.10 that ∑ ∞n=1 x n is convergentwhenever ∑ ∞n=1 ‖x n‖ 2 is finite.Lemma A.6.11. ⊕ ∞ n=1U n is a closed subspace of H, and ⊕ ∞ n=1U n = span ∪ ∞ n=1 U n .n=1Proof. We first prove that ⊕ ∞ n=1U n is closed. From the second assertion of the lemma,it will follow that ⊕ ∞ n=1U n is a subspace.Let x n ∈ ⊕ ∞ n=1U n with x n = ∑ ∞k=1 xk n, and assume that x n converges to x. We need toprove x ∈ ⊕ ∞ n=1U n . From Lemma A.6.10, ∑ ∞k=1 ‖xk n‖ 2 is finite for any n. In particular,by the triangle inequality and the relation (x + y) 2 ≤ 2x 2 + 2y 2 , ∑ ∞k=1 ‖xk n − x k m‖ 2 isfinite for any n, m. This implies that ∑ ∞k=1 xk n − x k m is convergent, and we obtain2‖x n − x m ‖ 2 ∞∑=x k n − x k ∞∑ ∥m= ∥x k n − x k ∥ 2 m .∥∥k=1Now, since x n is convergent, it is in particular a cauchy sequence. From the aboveequality, we then obtain that x k n is a cauchy sequence for every k, convergent to a limitx k . Since U k is closed, x k ∈ U k . We want to prove x = ∑ ∞k=1 xk . To this end, defineα n = (‖x k n‖) k≥1 . α n is then a member of l 2 , and‖α n − α m ‖ 2 l = ∑∞ ∞∑(‖x k n‖ − ‖x k m‖) 2 ≤ ‖x k 2n − x k m‖ 2 = ‖x n − x m ‖ 2 ,k=1so α n is cauchy in l 2 , therefore convergent to a limit α. In particular, αn k convergesto α k . But αn k = ‖x k n‖, and ‖x k n‖ converges to ‖x k ‖. Therefore, α k = ‖x k ‖. We maynow concludelimn∞ ∑k=1k=1k=1(‖x k n‖ − ‖x k ‖) 2 = limn‖α n − α‖ 2 l 2 = 0,

A.6 Hilbert Spaces 263in particular lim n∑ ∞k=1 ‖xk n‖ 2 = ∑ ∞k=1 ‖xk ‖ 2 . We then obtain by orthogonality,∥ n ∥ x − ∑ ∥∥∥∥2x kk=1∥ ∥∥∥∥ ∑∞= lim x k m −mk=1∥n∑ ∥∥∥∥2x kk=1∑n= lim ‖x k m − x k ‖ 2 +mk=1= limm∞∑=∞∑k=n+1k=n+1‖x k ‖ 2 ,‖x k m‖ 2∞∑k=n+1‖x k m‖ 2and since ∑ ∞k=1 ‖xk ‖ 2 is finite, it follows that x = ∑ ∞k=1 xk . Then x ∈ ⊕ ∞ n=1U n , andwe may finally conclude that ⊕ ∞ n=1U n is closed.Next, we need to prove the equality ⊕ ∞ n=1U n = span ∪ ∞ n=1 U n . As in the proof ofLemma A.6.9, it is clear that ⊕ ∞ n=1U n ⊆ span ∪ ∞ n=1 U n . To prove the other inclusion,note that we clearly have span ∪ ∞ n=1 U n ⊆ ⊕ ∞ n=1U n . Since we have seen that ⊕ ∞ n=1U nis closed, it follows that span ∪ ∞ n=1 U n ⊆ ⊕ ∞ n=1U n .Lemma A.6.12. Let (U n ) be a sequence of closed orthogonal subspaces and let P n bethe orthogonal projection onto U n . For any x ∈ ⊕ ∞ n=1U n , it holds that x = ∑ ∞n=1 P nx.Proof. Since x ∈ ⊕ ∞ n=1U n , we know that x = ∑ ∞n=1 x n for some x n ∈ U n . Since P k iscontinuous and the subspaces are orthogonal, we findas desired.P k x =∞∑P k x n = x n ,n=1Lemma A.6.13. For any subset A, it holds that A ∩ A ⊥ = {0}.Proof. Let x ∈ A ∩ A ⊥ . We find ‖x‖ 2 = 〈x, x〉 = 0, showing x = 0.Lemma A.6.14. Let M and N be orthogonal closed subspaces. Assume x ∈ M ⊕ N.If x ∈ M ⊥ , then x ∈ N.

264 AnalysisProof. Since N and M are orthogonal, N ⊆ M ⊥ . Let x = y + z with y ∈ M andz ∈ N. Then y = x − z ∈ M ⊥ . Since also y ∈ M, Lemma A.6.13 yields y = 0 andtherefore x = z ∈ N, as desired.Lemma A.6.15. Let U, M and N be closed subspaces. Assume that U and M areorthogonal, and assume that U and N are orthogonal. If U ⊕ M = U ⊕ N, thenM = N.Proof. By symmetry, it will suffice to show M ⊆ N. Assume that x ∈ M. Thenx ∈ U ⊕ M, and therefore x ∈ U ⊕ N. Now, since M ⊆ U ⊥ , we have x ∈ U ⊥ . ByLemma A.6.14, x ∈ N.Lemma A.6.16. Let (U n ) be a sequence of closed orthogonal subspaces. It then holdsfor any n ≥ 1 that(∞⊕n⊕) ( ∞)⊕U k = U k ⊕ U k .k=1k=1k=n+1Proof. It is clear that (⊕ n k=1 U k)⊕(⊕ ∞ k=n+1 U k) ⊆ ⊕ ∞ k=1 U k. To show the other inclusion,let x ∈ ⊕ ∞ k=1 U k. We then have x = ∑ ∞k=1 x k for some x k ∈ U k , where ∑ ∞k=1 ‖x k‖ 2 .Therefore, in particular ∑ ∞k=n+1 ‖x k‖ 2 . Thus, by the decompositionwe obtain the desired result.( n) (∑ ∑ ∞x = x k +k=1k=n+1x k),Lemma A.6.17. Let x, x ′ ∈ H and assume that 〈y, x〉 = 〈y, x ′ 〉 for all y ∈ H. Thenx = x ′ .Proof. It follows that x − x ′ is orthogonal to H. From Lemma A.6.13, x − x ′ = 0 andtherefore x = x ′ .Lemma A.6.18. Let H be a Hilbert space and let U be a closed subspace. Let Pdenote the orthogonal projection onto U. Let x ∈ H and let x ′ ∈ U. If it holds for ally ∈ U that 〈y, x〉 = 〈y, x ′ 〉, then x ′ = P x.

A.6 Hilbert Spaces 265Proof. Let Q be the orthogonal projection onto U ⊥ . We then have, for any y ∈ U,〈y, P x〉 = 〈y, Qx + P x〉= 〈y, x〉= 〈y, x ′ 〉.Since P x and x ′ are both elements of U, and U is a Hilbert space since it is a closedsubspace of H, we conclude by Lemma A.6.17 that P x = x ′ .We end with a few results on weak convegence. We say that a sequence x n is weaklyconvergent to x if 〈y, x n 〉 tends to 〈y, x〉 for all y ∈ H.Lemma A.6.19. If x n has a weak limit, it is unique.Proof. See Lemma 3.6.3 of Hansen (2006).Lemma A.6.20. If x n converges to x, it also converges weakly to x.Proof. This follows immediately by the continuity properties of the inner product.Theorem A.6.21. Let H be a Hilbert space and let (x n ) be a bounded sequence in H.Then (x n ) has a weakly convergent subsequence.Proof. See Schechter (2002), Theorem 8.16, where the theorem is proved for reflexiveBanach spaces. Since every Hilbert space is reflexive according to Corollary 11.10 ofMeise & Vogt (1997), this proves the claim.Lemma A.6.22. Assume that (x n ) converges weakly to x. Then ‖x‖ ≤ lim inf ‖x n ‖.Proof. See Theorem 3.4.11 of Ash (1972).Lemma A.6.23. Let x n be a sequence in H converging weakly to x. Let A : H → Hbe linear and continuous. Then Ax n converges weakly as well.Proof. This follows from Theorem 3.4.11 of Ash (1972).

266 AnalysisA.7 Closable operatorsIn this section, we consider the concept of closable and closed operators. This will beused when extending the Malliavin derivative. Let X and Y be Banach spaces, thatis, complete normed linear spaces. Consider a subspace D(A) of X and an operatorA : D(A) → Y. We say that A is closable if it holds that whenever (x n ) is a sequencein D(A) converging to zero and Ax n is convergent, then the limit of Ax n is zero. Wesay that A is closed if it holds that whenever (x n ) is a sequence in D(A) such thatboth x n and Ax n are convergent, then lim x n ∈ D(A) and A(lim x n ) = lim Ax n .Lemma A.7.1. Let A be some operator with domain D(A). A is closed if and onlyif the graph of A is closed.Proof. This follows since the graph of A is closed precisely if it holds for any convergentsequence (x n , Ax n ) where x n ∈ D(A) that lim x n ∈ D(A) and A(lim x n ) = lim Ax n .Lemma A.7.2. Let A be a closable operator. We define the domain D(A) of theclosure of A as the set of x ∈ X such that there exists x n in D(A) converging to x andsuch that Ax n is convergent. The limit of Ax n is independent of the sequence x n andD(A) is a linear space.Proof. We first prove the result on the uniqueness of the limit. Assume that x ∈ D(A)and consider two different sequences x n and y n in D(A) converging to x such thatAx n and Ay n are both convergent. We need to prove that the limits are the same.But x n − y n converges to zero and A(x n − y n ) is convergent, so since A is closable, itfollows that A(x n − y n ) converges to zero and thus lim n Ax n = lim n Ay n , as desired.Next, we prove that D(A) is a linear space. Let x, y ∈ D(A) and let λ, µ ∈ R. Let x nand y n be sequences in D(A) converging to x and y, respectively. Then λx n + µy n isa sequence in D(A) converging to λx + µy, and let A(λx n + µy n ) is convergent sinceA is linear. Thus, λx + µy ∈ D(A) and D(A) is a linear space.Theorem A.7.3. Assume that A is closable. There is a unique extension A of A asa linear operator from D(A) to D(A), and the extension is closed.Proof. Let x ∈ D(A). From Lemma A.7.2, we know that there is x n in D(A) convergingto x and such that Ax n is convergent, and the limit of Ax n is independent of x n .

A.7 Closable operators 267We can then define Ax as the limit of Ax n . We need to prove that this extension islinear and closed.To prove linearity, let x, y ∈ D(A) and let λ, µ ∈ R. Let x n and y n be sequences inD(A) converging to x and y, respectively, such that Ax n and Ay n converges to Axand Ay, respectively. Then λx n + µy n converges to λx + µy and A(λx n + µy n ) isconvergent. By definition, the limit is A(λx + µy), and we concludeA(λx + µy) = limnA(λx n + µy n ) = limnλAx n + µAy n = λAx + µAy,as desired. To show that the extension is closed, we show that the graph of A is theclosure of the graph of A. Assume that (x, y) is a limit point of the graph of A, thenthere exists x n in D(A) such that x = lim x n and y = Ax n . By definition, we obtainx ∈ D(A) and Ax = y. Therefore, (x, y) is in the graph of A. On the other hand, if(x, y) is in the graph of A, in particular x ∈ D(A) and Ax = y. Then, there existsx n in D(A) converging to x such that Ax n is convergent as well, and the limit is Ax.Thus, (x n , Ax n ) converges to (x, Ax), so (x, y) is a limit point of the graph of A. Weconclude that the graph of A is the closure of the graph of A, and by Lemma A.7.1,A is closed.

268 Analysis

Appendix BMeasure theoryThis appendix contains some results from abstract measure theory and probabilitytheory which we will need. Our main sources are Hansen (2004a), Berg & Madsen(2001), Jacobsen (2003) and Rogers & Williams (2000a).B.1 The Monotone Class TheoremIn this section, we state the monotone class theorem and develop an important corollary,used when analysing the Brownian motion under the usual conditions.Definition B.1.1. A monotone vector space H on Ω is a subspace of the vector spaceB(Ω) of bounded real functions on Ω, such that H contains all constant functions andsuch that if (f n ) is a sequence of nonnegative functions increasing pointwise towardsa bounded function f, then f ∈ H.Theorem B.1.2 (Monotone Class Theorem). Let K ⊆ B(Ω) be stable undermultiplication, and let H ⊆ B(Ω) be a monotone vector space. If K ⊆ H, H containsall σ(K)-measurable bounded functions.Proof. The theorem is stated and proved as Theorem I.21 in Dellacherie & Meyer(1975), under the assumption that H is closed under uniform convergence. Since

270 Measure theorySharpe (1988) proves that any monotone vector space is closed under uniform convergence,the claim follows.By C([0, ∞), R n ), we denote the continuous mappings from [0, ∞) to R n . We endowthis space with the σ-algebra C([0, ∞), R n ) induced by the coordinate projections X ◦ t ,X ◦ t (f) = f(t) for any f ∈ C([0, ∞), R n ).Lemma B.1.3. Let F be closed in R n . There exists (f n ) ⊆ Cc∞converges pointwise to 1 F .(R n ) such that f nProof. By Cohn (1980), Proposition 7.1.4, there exists a sequence G n of open setssuch that F ⊆ G n and F = ∩ ∞ n=1G n . Let K n be an increasing sequence of compactsets with ∪ ∞ n=1K n . Then F ∩ K n is compact, so by Theorem A.3.4, there is f n withF ∩ K n ≺ f n ≺ G n . We claim that this sequence of functions fulfills the conditionsin the lemma. Let x ∈ F . From a certain point onwards, x ∈ F ∩ K n , and thenf n (x) = 1. Assume next x /∈ F . Then, x /∈ G n from a certain point onwards, and thenf n (x) = 0. We conclude lim f n = 1 F .Corollary B.1.4. Let H ⊆ B(C([0, ∞, R n )) be a monotone vector space. If H containsall functions of the form ∏ nk=1 f k(X ◦ t k) with f k ∈ C c (R n ), then H contains allC([0, ∞), R n ) measurable functions.Proof. Since the compact sets are closed under finite intersections, it is clear thatC c (R n ) is an algebra. Therefore, putting{ n}∏K = f k (Xt ◦ k)∣ 0 ≤ t 1 ≤ · · · ≤ t n and f 1 , . . . , f n ∈ C c (R) ,k=1it is clear that K is an algebra. By Theorem B.1.2, H contains all σ(K) meausrablemappings. We wish to argue that K = C([0, ∞), R n ). It is clear that K ⊆ C([0, ∞), R n ),so it will suffice to show the other inclusion. To do so, we only need to show that allcoordinate projections are K meausrable. Let t ≥ 0 and let F be closed in R n . FromLemma B.1.3, there exists a sequence (f n ) ⊆ Cc ∞ (R n ) converging pointwise to 1 F .This means that 1 F (Xt ◦ ) is the pointwise limit of f n (Xt ◦ ), so 1 F (Xt ◦ ) is K-measurable.This shows that (Xt ◦ ∈ F ) ∈ K. Therefore Xt◦ is K measurable, as desired.

B.2 L p and almost sure convergence 271B.2 L p and almost sure convergenceThis section contains some of the classical results on L p and almost sure convergence.Most of these results are surely well-known, we repeat them here for completeness.Lemma B.2.1. Let (E, E, µ) be a measure space. Assume that f n converges to f andg n converges to g in L 2 (E). Then f n g n converges to fg in L 1 (E).Proof. By the Cauchy-Schwartz inequality,‖f n g n − fg‖ 1 ≤ ‖f n g n − f n g‖ 1 + ‖f n g − fg‖ 1so since ‖f n ‖ 2 is bounded, the conclusion follows.≤ ‖f n ‖ 2 ‖‖g n − g‖ 2 + ‖f n − f‖ 2 ‖g‖ 2 ,Lemma B.2.2. Let (E, E, µ) be a measure space. Assume that f n converges in L 1 tof, and assume that g n converges almost surely to g. If the sequence g n is dominatedby a constant, then f n g n converges in L 1 to fg.Proof. We have‖f n g n − fg‖ 1 ≤ ‖f n g n − fg n ‖ 1 + ‖fg n − fg‖≤ M‖f n − f‖ 1 + ‖fg n − fg‖ 1 .Here, the first term trivially tends to zero. By dominated convergence with bound2Mf, the second term also tends to zero.Lemma B.2.3 (Scheffé). Let (E, E, µ) be a measure space and let f n be a sequence ofprobability densities with respect to µ. Let f be another probability density with respectto µ. If f n converges µ-almost surely to f, then f n converges to f in L 1 .Proof. This is Lemma 22.4 of Hansen (2004a).Lemma B.2.4. Let (E, E, µ) be a finite measure space. If 1 ≤ p ≤ r, it holds thatL r (E) ⊆ L p (E) and L r convergence implies L p convergence.Proof. See Theorem 7.11 of Berg & Madsen (2001).

272 Measure theoryLemma B.2.5. Let (E, E, µ) be a measure space. If f n converges to f in L p , there isa subsequence f nk converging µ almost surely to f, dominated by a mapping in L p .Proof. This is proved as Corollary 7.20 in Berg & Madsen (2001).B.3 Convergence of normal distributionsIn this section, we prove a result on convergence of normal distributions which will beessential to our development of the Malliavin calculus. We begin with a few lemmas.Lemma B.3.1 (Hölder’s inequality). Let (E, E, µ) be a measure space and letf 1 , . . . , f n be real measurable mappings for some n ≥ 2. Let p 1 , . . . , p n ∈ (1, ∞) with∑ nk=1 1p k= 1. It then holds that ‖ ∏ nk=1 f k‖ 1 ≤ ∏ nk=1 ‖f k‖ pn . We say that p 1 , . . . , p nare conjugate exponents.Proof. We use induction. The case n = 2 is just the ordinary Hölder’s inequality,proving the induction start. Assume that the results holds in the case of n, we willprove it for n + 1. Let q = p n+1p , then q is the conjugate exponent to p n+1−1 n+1, so theordinary Hölder’s inequality yields ‖ ∏ n+1k=1 f k‖ ≤ ‖ ∏ nk=1 f k‖ q‖f n+1 ‖ pn+1 . Next, notethat ∑ nk=1 1p k= 1 − 1p n+1= pn+1−1p n+1= 1 qthe conjugate exponents p 1q, . . . , p nq, we obtain∥ n∏ ∥∥∥∥ f∥ kk=1 q=≤==, so applying the induction hypothesis with(∥ ∥ ∥∥∥∥ ∏n ∥∥∥∥|f k | qk=1 1(∏ nk=1(∏ n (∫k=1) 1q‖|f k | q ‖ p 1q) 1qn∏‖f k ‖ pk ,k=1) q ) 1qp 1|f k | p1 dµwhich, combined with our earlier finding, yields the desired result.

B.3 Convergence of normal distributions 273Lemma B.3.2. Let X be a normal distribution with mean µ and variance σ 2 . Letn ∈ N 0 . Then we haveEX n =[n/2]∑k=0(2k)!2 k k! σ2k µ n−2k .Proof. Since the odd central moments of X are zero, we obtainas desired.EX n = E(X − µ + µ) nn∑( k= E(X − µ)n)k µ n−k==k=0[n/2]∑k=0[n/2]∑k=0( 2kn)E(X − µ) 2k µ n−2k(2k)!2 k k! σ2k µ n−2k ,Lemma B.3.3. Let X k be a sequence of n-dimensional normally distributed variables.Assume that for any i ≤ n, X i k converges weakly to X. Let f Rn to R have polynomialgrowth. Then f(X k ) is bounded over k in L p for any p ≥ 1.Proof. If f has polynomial growth, then so does |f| p for any p ≥ 1. Therefore, itwill suffice to show that f(X k ) is bounded in L 1 for any f with polynomial growth.Let p be a polynomial in n variables dominating f. Assume that p has degree mand p(x) = ∑ |a|≤m λ ∏ na i=1 xai i , using usual multi-index notation. Obviously, we thenobtain|f(X k )| ≤ |p(X k )|≤∑ n∏|λ a | |(Xk) i a i|≤|a|≤m∑|a|≤m|λ a |i=1n∏(1 + (Xk) i 2 ) ai ).Letting q(x) = ∑ |a|≤m |λ a| ∏ ni=1 (1 + x2 i )ai , q is a polynomial of degree 2m in n variables,and we have E|f(X k )| ≤ E|q(X k )|. Furthermore, the exponents in q are alleven. Thus, it will suffice to show the result in the case f is a polynomial with evenexponents.i=1

274 Measure theoryTherefore, assume that f is a polynomial in n variables of degree m. Assume thatwe have f(x) = ∑ a∈I λ ∏ n n am i=1 xa ii , where λ a is zero whenever a has an odd element.The generalized Hölder’s inequality of Lemma B.3.1 then yieldsE|f(X k )| ≤ ∑ ( n)∏|λ a |E (Xk) i a i≤|a|≤m∑|a|≤m|λ a |i=1n∏ (E(Xik ) na ) 1 i n.Let µ i k denote the mean of Xi k and σi k denote the standard deviation of Xi k . SinceXk i is normally distributed and converges weakly, µi k and σi kare convergent, thereforebounded. Now, by Lemma B.3.2, E(Xk i )n is a continuous function of µ i k and σi k forfixed i and n. Therefore, k ↦→ E(Xk i )n is bounded for any i and any n. This showsthat E|f(X k )| is bounded over k, as was to be proved.Theorem B.3.4. Let X k be a sequence of n-dimensional variables converging in probabilityto X. Assume that (X k , X) is normally distributed for any k. Let f ∈ C 1 p(R n ).Then f(X k ) converges to f(X) in L p , p ≥ 1.i=1Proof. Let f ∈ Cp(R 1 n ) be given. Let p i be a polynomial in n variables dominating∂f∂x i. The mean value theorem yields, for some ξ k on the line segment between X k andX,n∑|f(X k ) − f(X)| =∂f∣ (ξ k )(Xk i − X i )∂x i∣≤i=1n∑|p i (ξ k )||Xk i − X i |.i=1Now, defining Yk i = 2 + (Xi k )2 + (X i ) 2 , we have |ξk i | ≤ |Xi k | + |Xi | ≤ Yk i and therefore|p i (ξ k )| ≤ q i (Y k ) for some other polynomial q i . Using Jensen’s inequality and theCauchy-Schwartz inequality, we obtain( n) p∑E|f(X k ) − f(X)| p ≤ E q i (Y k )|Xk i − X i |k=1∑n≤ En p (qi (Y k )|Xk i − X i | ) pi=1∑n= n p Eq i (Y k ) p |Xk i − X i | pi=1∑n≤ n p ‖q i (Y k ) p ‖ 2 ‖|Xk i − X i | p ‖ 2 .i=1

B.4 Separability and bases for L 2 -spaces 275Now, q i (Y k ) is a polynomial transformation of the 2n-dimensional normally distributedvariable (X k , X). Since X k converges in probability to X k , we also have weak convergence.Thus, (X k , X) converges coordinatewisely weakly to (X, X). By Lemma B.3.3,we may then conclude that ‖q i (Y k ) p ‖ 2 is bounded over k for any i ≤ n. Let C be abound, we can then conclude∑nE|f(X k ) − f(X)| p ≤ Cn p ‖|Xk i − X i | p ‖ 2i=1∑n= Cn p (E|Xik − X i | 2p) 1 2.To prove the desired result, it will therefore suffice to prove that Xk i tends to Xi inL p for any p ≥ 1. It will be enough to prove that we have L 2n convergence for n ∈ N.Let n ∈ N be given. Since Xk i converges in probability to Xi , Xk i − Xi converges inprobability to zero, so Xk i − Xi converges in distribution to the point measure at zero.Because we have asumed that (X k , X) is normally distribued, Xk i − Xi is normallydistributed. Let ξk i be the mean and let νi kbe the standard deviation. We thenconcluce that ξk i and νi kboth tend to zero as k tends to infinity. By Lemma B.3.2, wethen conclude that E(Xk i − Xi ) 2n tends to zero, showing the desired convergence inL 2n and implying the result of the lemma.i=1B.4 Separability and bases for L 2 -spacesThis section yields some results on L 2 spaces which will be necessary for the developmentof the Hilbert space results of the Malliavin calculus.Lemma B.4.1. If (E, E, µ) is a finite measure space which is countably generated,then L 2 (E) is separable as a pseudometric space.Proof. Let D be a countable generating family for E. We can assume without lossof generality that E ∈ D. Let H denote the family of sets ∩ n k=1 A k where A k ∈ Dfor k ≤ n. Then H is countable, D ⊆ H, and H is stable under intersections. PutS = {1 A |A ∈ E} and S H = {1 A |A ∈ H}.Step 1: span S H is dense in L 2 (E, E, µ). We will begin by showing that span S H isdense in L 2 (E). After doing so, we will identify a dense subset of span S H . To showthat span S H is dense, first note that from Theorem 7.27 of Berg & Madsen (2001),

276 Measure theoryspan S is dense in L 2 (E). Therefore, to show that span S H is dense in L 2 (E), it willsuffice to show that span S H is dense in span S. And to do so, it will suffice to showthat S ⊆ span S H .Define K = {A ∈ E|A ∈ span S H }. To show that S ⊆ span S H , we must show thatK = E. Clearly, H ⊆ K. Since H is a generating family for E stable under intersections,it will suffice to show that K is a Dynkin class.Clearly, E ∈ K. Assume that A, B ∈ K with B ⊆ A. Then 1 A\B = 1 A − 1 B . Sinceeach of these is in span S H , so is 1 A − 1 B and therefore A \ B ∈ K. Finally, assumethat A n is an increasing sequence in K and put A = ∪ ∞ n=1A n . Since µ is finite, 1 Antends to 1 A in L 2 (E) by dominated convergence. Since 1 An ∈ span S H for all n andspan S H is closed, 1 A ∈ span S H and therefore A ∈ K. Thus, K is a Dynkin class. Weconclude that K = E and therefore S ⊆ span S H , from which we deduce that span S His dense in L 2 (E).Step 2: Identification of a dense countable set. In order to show separabilityof L 2 (E), it will suffice to show that there exists a dense countable subset of span S H .Define{ n∣ }∑ ∣∣∣∣V = λ k 1 Ak λ k ∈ Q, A k ∈ H, k ≤ n .k=1Since H is countable, V is countable. We will show that V is dense in span S H . Letf ∈ span S H be given, f = ∑ nk=1 λ k1 Ak . Let λ j k tend to λ k through Q and definef j = ∑ nk=1 λj k 1 A k. Then f j ∈ V andn∑‖f − f j ‖ 2 ≤ |λ k − λ j k |‖1 A k‖ 2 ,showing the desired result.k=1Lemma B.4.2. Let (E, E, µ) and (K, K, ν) be two finite measure spaces which arecountably generated. Let ([f i ]) and ([g j ]) be countable orthonormal bases for L 2 (E) andL 2 (K), respectively. Then the family ([f i ⊗ g j ]) is an orthonormal basis for L 2 (E ⊗ K).Proof. We first show that the family is orthonormal. By the Tonelli theorem, we find‖f i ⊗ g j ‖ 2 = ‖f i ‖ 2 ‖g j ‖ 2 = 1. To show that the family is orthogonal, consider f i ⊗ g jand f k ⊗ g n . We then obtain by the Fubini theorem,∫〈f i ⊗ g j , f k ⊗ g n 〉 = f i (s)g j (t)f k (s)g n (t) d(µ ⊗ ν)(s, t)= 〈f i , f k 〉〈g j , g n 〉,

B.5 Uniform integrability 277showing that if i ≠ k or j ≠ n, f i ⊗ g j is orthogonal to f k ⊗ g n , as desired. It remainsto show that the family is complete. To this end, consider ψ ∈ L 2 (E ⊗ K) and assumethat ψ is orthogonal to f i ⊗ g j for any i and j. We need to show that ψ is almostsurely zero. To this end, we first calculate∫〈ψ, f i ⊗ g j 〉 = ψ(s, t)f i (s)g j (t) d(µ ⊗ ν)(s, t)=∫ ∫ψ(s, t)f i (s) dµ(s)g j (t) dν(t)=∫〈ψ(·, t), f i 〉g j (t) dν(t).Put φ i (t) = 〈ψ(·, t), f i 〉. By the Cauchy-Schwartz inequality and Fubini’s theorem, ψ iis in L 2 (K), and the above shows that 〈φ i , g j 〉 = 〈ψ, f i ⊗ g j 〉 = 0, so since ([g j ]) isan orthonormal basis of L 2 (K), we conclude that φ i is ν almost surely zero. Since acountable union of null sets again is a null set, we find that ν almost surely, φ i (t) = 0for all i. For any such t, ψ(·, t) is orthogonal to all f i , showing that ψ(·, t) is zero µalmost surely. All in all, we can then conclude that ψ is zero µ ⊗ ν almost surely, asdesired.B.5 Uniform integrabilityLet (X i ) i∈I be a family of stochastic variables. We say that X i is uniformly integrableiflim E|X i |1 (|Xi |>x) = 0.supx→∞ i∈IWe will review some basic results about uniform integrability. We refer the resultsmainly for discrete sequences of variables, but the result extend to sequences indexedby [0, ∞) as well.Lemma B.5.1. Let (X i ) i∈I be a family of stochastic variables. If (X i ) is bounded inL p for some p > 1, then (X i ) is uniformly integrable.Proof. See Lemma 20.5 of Rogers & Williams (2000a).Lemma B.5.2. Let (X i ) i∈I be uniformly integrable. Then (X i ) is bounded in L 1 .Proof. This follows from Lemma 20.7 of Rogers & Williams (2000a).

278 Measure theoryLemma B.5.3. Let X n be a sequence of stochastic variables, and let X be anothervariable. X n converges in L 1 to X if and only if X n is uniformly integrable andconverges in probability to X.Proof. This is Theorem 21.2 of Rogers & Williams (2000a).Lemma B.5.4. Let X n be a sequence of nonnegative integrable variables, and let Xbe another integrable variable. Assume that X n converges almost surely to X. (X n ) isuniformly integrable if and only if EX n converges to EX.Proof. First assume that (X n ) is uniformly integrable. Since X n also converges inprobability to X, it follows from Lemma B.5.3 that X n converges in L 1 to X, andtherefore the means converge as well.Conversely, assume that EX n converges to EX. We have|X n − X| = (X n − X) + + (X n − X) −= X n − X + (X n − X) − + (X n − X) −= X n − X + 2(X n − X) − .Now, since X n converges to X almost surely, X n − X converges to 0 almost surely.Because x ↦→ x − is continuous, (X n − X) − converges to 0 almost surely. Clearly,X must be nonnegative. Thus, both X n and X are nonnegative. Since x ↦→ x − isdecreasing, this yields (X n −X) − ≤ (−X) − = X. By dominated convergence, we thenobtainlimnE|X n − X| = limnEX n − EX + 2 limnE(X n − X) − = 2E limn(X n − X) − = 0.B.6 MiscellaneousThis final section of the appendix contains a variety of results which do not really fitanywhere else. We begin with some results on density of a certain class of stochasticvariables in L 2 spaces.

B.6 Miscellaneous 279Lemma B.6.1. Let (E, E, µ) be a measure space and let f 1 , . . . , f n be real mappingson E. If f 1 , . . . , f n has exponential moments of all orders, then so does ∑ nk=1 λ kf k ,where λ ∈ R n .Proof. We proceed by induction. For n = 1, there is nothing to prove. Assume thatwe have proved the claim for n. We find for λ ∈ R by the Cauchy-Schwartz inequalitythat∫ ( )n+1∑exp λ λ k f k dµ=≤∫k=1( n)∑exp λλ k f k exp (λλ n+1 f n+1 ) dµk=1( ∫exp(2λ)n∑λ k f k dµk=1) 12 (∫which is finite by the induction hypothesis, proving the claim.) 12exp (2λλ n+1 f n+1 ) dµ ,Theorem B.6.2. Let (E, E, µ) be a measure space, where µ is a finite measure. Assumethat E is generated by a family of real mappings K, where all mappings in Khave exponential moments of all orders. Then the variables( n)∑exp λ k f k ,k=1where f 1 , . . . , f n ∈ K and λ ∈ R n , are dense in L 2 (E).Proof. Let H denote the family of variables of the form exp ( ∑ nk=1 λ kf k ). Note thatby Lemma B.6.1, all elements of H have moments of all orders, so H is a subset ofL 2 (E) and the conclusion of the theorem is well-defined.Now, let g ∈ L 2 (E) be given, orthogonal to H. We wish to show that g is µ almostsurely equal to zero. Consider f 1 , . . . , f n ∈ K. Since g ∈ L 2 (E) and µ is finite, g ∈L 1 (E), so g · µ is a well-defined signed measure. Define ν = (f 1 , . . . , f n )(g · µ), then ν isalso a signed measure, and for any A ∈ B k , we then have ν(A) = ∫ 1 A (f 1 , . . . , f n )g dµ.The Laplace transform of ν is∫ ( n)∑∫ ( n)∑exp λ k x k dν(x) = exp λ k f k d(g · µ)k=1=∫k=1( n)∑exp λ k f k g dµ,k=1

280 Measure theoryand the last expression is equal to zero by assumption. Thus, ν has Laplace transformidentically equal to zero, so by Lemma B.6.6, ν is the zero measure. That is,∫1A (f 1 , . . . , f n )g dµ for any A ∈ B k . Since the sets ((f 1 , . . . , f n ) ∈ A) is stable underintersections and generates E, the Dynkin Lemma yields that ∫ 1 G g dµ for any G ∈ E.This shows that g is µ almost surely zero.Next, we review some basic results on measurability on product spaces.Lemma B.6.3. Let (E, E) and (K, K) be measurable spaces. Let E and K be generatingsystems for E and K, respectively, closed under intersections. If E contains E and Kcontains K, then E ⊗ K is generated by the sets of the form A × B, where A ∈ E andB ∈ K.Proof. Let H be the σ-algebra generated by the sets of the form A × B, where A ∈ Eand B ∈ K. We need to prove E ⊗ K = H. Clearly, H ⊆ E ⊗ K, we need to prove theother inclusion. To do so, note that letting D be the family of sets A × B where A ∈ Eand B ∈ K, E ⊗ K is generated by D. It will therefore suffice to prove D ⊆ H.To this end, let F be the family of sets B ∈ K such that E × B ∈ H. Then F is stableunder complements and increasing unions, and since E ∈ E, K ⊆ F. In particular,K ∈ F. We conclude that F is a Dynkin class containing K, therefore K ⊆ F. Thisshows that E × B ∈ H for any B ∈ K. Analogously, we can prove that A × K ∈ H forany A ∈ E. Letting A ∈ E and B ∈ K, we then obtain A×B = (A×K)∩(E ×B) ∈ H,as desired. We conclude D ⊆ H and therefore H = E ⊗ K.Lemma B.6.4. Let (E, E) and (K, K) be measurable spaces. Let E and K be classesof real functions on E and K, respectively. Assume that there exists sequences offunctions in E and K, respectively, converging towards a nonzero constant. Assumethat E = σ(E) and K = σ(K). The σ-algebra E ⊗ K is generated by the functions onthe form f ⊗ g, where f ∈ E and g ∈ K.Proof. Let H be the σ-algebra generated by the functions f ⊗ g, where f ∈ E andg ∈ K. Clearly, H ⊆ E ⊗ K. We need to show the other inclusion. By definition, f ⊗ gis H-measurable whenever f ∈ E and g ∈ K. Now, let f ∈ E be fixed. By assumption,there is a sequence g n in K converging towards a constant, say c. Then we find thatcf(x) = c(f(x) lim g n (y)) = lim c(f ⊗ g n )(x, y),

B.6 Miscellaneous 281showing that the function (x, y) ↦→ f(x) is H-measurable for any f ∈ E. This meansthat f −1 (A) × K ∈ H for any f ∈ E and A ∈ B. Now, the sets {B ∈ E|B × K ∈ H}form a σ-algebra, and by what we have just shown, this σ-algebra contains the family{f −1 (A)|f ∈ E, A ∈ B}, which generates E. We can therefore conclude that B×K ∈ Hfor any B ∈ E. In the samme manner, we can conclude that E ×C ∈ H for any C ∈ K.Therefore, for B ∈ E and C ∈ K, we findB × C = (B × K) ∩ (E × C) ∈ H.These sets generate E ⊗ K, and we may therefore conclude that E ⊗ K ⊆ H.Lemma B.6.5. Let (E, E, µ) and (K, K, ν) be two measure spaces. Let f n be a sequenceof real mappings on E × K, measurable with respect to E ⊗ K. Let p ≥ 1 andassume that f n tends to f in L p (E ⊗ K). There is a subsequence such that for ν almostall y, ∫ |f nk (x, y)| p dµ(x) tends to ∫ |f(x, y)| p dµ(x).Proof. Since ∫ |f n − f| p d(µ ⊗ ν) tends to zero, the mapping in y on K given by∫|fn (x, y) − f(x, y)| p dµ(x) tends to zero in L 1 (K). By Lemma B.2.5, there is asubsequence converging ν almost surely. Let y be an element of K such that we haveconvergence. By the inverse triangle inequality of the norm in L p (E), we obtain(∫∣) 1 (∫|f nk (x, y)| p pdµ(x) −|f(x, y)| p dµ(x)) 1p∫∣ ≤ |f nk (x, y) − f(x, y)| p dµ(x),showing that ∫ |f nk (x, y)| p dµ(x) tends to ∫ |f(x, y)| p dµ(x), as desired.Finally, a few assorted results.Lemma B.6.6. Let µ and ν be two bounded signed measures on R ktransforms ψ and ϕ, respectively. If ψ = ϕ, then µ = ν.with LaplaceProof. See Jensen (1992), Theorem C.1.Lemma B.6.7 (Doob-Dynkin lemma). Let X be a variable on (Ω, F, P ) takingvalues in a polish space S. Let Y be another variable on the same probability spacetaking its values in a measurable space (E, E). X is σ(Y ) measurable if and only ifthere exists a measurable mapping ψ : E → S such that X = ψ(Y ).Proof. See Dellacherie & Meyer (1975), Theorem 18 of Section 12.1. The originalproof can be found in Doob (1953), p. 603.

282 Measure theoryLemma B.6.8. Let (E, E, µ) be a measure space and let K be a sub-σ-algebra of E.Let f be K measurable. With µ |K denoting the restriction of µ to K, we have∫ ∫f dµ = f dµ |K .Proof. If A ∈ K, we clearly have∫∫1 A dµ = µ(A) = µ |K (A) =1 A dµ |K .By linearity, the result extends to simple K measurable mappings. By monotoneconvergence, it extends to nonnegative K measurable mappings. By splitting intopositive and negative parts, the proof is finished.Lemma B.6.9. Let (Ω, F, P ) be a probability space. Let N be the null sets of thespace and define G = σ(F, N ). Any G measurable real variable is almost surely equalto some F measurable variable.Proof. Let X be G measurable. Let ξ be a version of X − E(X|F). If we can provethat ξ is almost surely zero, we are done. Note that with A denoting the almost suresets of F, G = σ(F, A). Since both F and A contains Ω, the sets of the form A ∩ Bwhere A ∈ F and B ∈ A form a generating system for G stable under intersections.By the Dynkin lemma, to prove that ξ is almost surely zero, it will suffice to provethat E1 A∩B ξ for A ∈ F and B ∈ N . To this end, we simply rewriteE1 A∩B ξ)E1 A ξ) = E1 A (X − E(X|F)) = E1 A X − EE(1 A X|F) = 0.The lemma is proved.

Appendix CAuxiliary resultsThis appendix contains some results which were developed in connection with the mainparts of the thesis, but which eventually turned out not to fit in or became replacedby simpler arguments. Since the results have independent interest, their proofs arepresented here.C.1 A proof for the existence of [M]In this section, we will discuss the existence of the quadratic variation for a continuousbounded martingale. This existence is proved in several different ways in the litteratur.Protter (2005) defines the quadratic variation directly through an integration-by-partsformula and derives the properties of the quadratic variation afterwards. Karatzas &Shreve (1988) uses the Doob-Meyer Theorem to obtain the quadratic variation as thecompensator of the squared martingale. The most direct proof seems to be the onefound in Rogers & Williams (2000b). This proof, however, still requires a substantialbackground, including preliminary results on stochastic integration with respect toelementary processes. Furthermore, even though the proof may seem short, a gooddeal of details are omitted, and the full proof is actually very lengthy and complicated,as may be seen in, say, Jacobsen (1989) or Sokol (2005), including a good deal ofcumbersome calculations.

284 Auxiliary resultsWe will here present an alternative proof. The proof is a bit longer, but its structure issimpler, using quite elementary theory and requiring few prerequisites. In particular,no development of the stochastic integral is needed for the proof. The virtue of thisis that it allows the development of the quadratic variation before the developmentof the stochastic integral. This yields a separation of concerns and therefore a moretransparent structure of the theory. Before beginning the proof, we will outline themain ideas and the progression of the section.In the remainder of the section, we work in the context of a filtered probability space(Ω, F, P, F t ) and a continuous bounded martingale M.Consider a partition π of [0, t] and let s ≤ t. By π s , we denote the partition of [0, s]given by π s = {s}∪π ∩[0, s]. Our plan is to show that the limit of the process given ass ↦→ Q π s(M) converges uniformly in L 2 on [0, t] as the partition grows finer. We willthen define the limit as the quadratic variation of M. Demonstrating this convergencewill require two main lemmas, the relatively easy Lemma C.1.3 and the somewhatdifficult Lemma C.1.4. After having defined the quadratic variation in this manner,we will argue that it is continuous and increasing and show that it can alternativelybe characterised as the unique increasing process such that M 2 − [M] is a uniformlyintegrable martingale. We will show how to extend the definition of the quadraticvariation to continuous local martingales and show how to characterise the quadraticvariation when M is a square-integrable martingale and when M is a continuous localmartingale.We begin with a few definitions concerning convergence of sequences indexed by partitions.Let t ≥ 0, and let (x π ) be a family of elements of a pseudometric space (S, d),indexed by the partitions of [0, t]. We call (x π ) a net, inspired by the usual definitionof nets as given in, say, Munkres (2000).We say that x π is convergent to x if it holds that for any ε > 0, there exists a partitionπ such that whenever ϖ ⊇ π is another partition, d(x ϖ , x) < ε. We say that x π iscauchy if it holds that for any ε > 0, there exists a partition π such that wheneverϖ, ϑ ⊇ π are two other partitions, d(x ϖ , x ϑ ) < ε. Clearly, to show that a net is cauchy,it will suffice to show that for any ε > 0, there exists a partition such that wheneverϖ ⊇ π, d(x ϖ , x π ) ≤ ε.Lemma C.1.1. Let x π be a net in (S, d). If x π has a limit, it is unique. If (S, d) iscomplete and x π is cauchy, then x π has a limit.

C.1 A proof for the existence of [M] 285Proof. Assume that x π has limits x and y. For any partition π, it clearly holds thatd(x, y) ≤ d(x, x π ) + d(x π , y). From this, it immediately follows that limits are unique.Now assume that (S, d) is a complete pseudometric space and that x π is cauchy. Foreach n, let π n be the partition such that whenever ϖ, ϑ ⊇ π n , d(x ϖ , x ϑ ) ≤ 12. Bynexpanding the partitions, we can assume without loss of generality that π n is increasing.We then obtain d(x πn , x πn+1 ) ≤ 12, and in particular, for any n ≥ m,nd(x πn , x πm ) ≤m∑k=n+1d(x πk−1 , x πk ) ≤ 12 n .Therefore, x πn is a cauchy sequence, therefore convergent. Let x be the limit. Wewish to show that x is the limit of the net x π . By continuity of the metric, we obtaind(x πn , x) ≤ 12. Let ε > 0 be given. Let n be such that 1n 2< ε. Then we have, fornany ϖ ⊇ π n ,d(x ϖ , x) ≤ d(x ϖ , x πn ) + d(x πn , x) ≤ 22 n ≤ 2ε.Since ε was arbitrary, we have convergence.Lemma C.1.2. Let x π be convergent with limit x. Let ε n be a positive sequence tendingto zero. For each n, let π n be a partition such that whenever ϖ ⊇ π n , d(x ϖ , x) ≤ ε n .Then x πn is a sequence converging to x.Proof. This is immediate.The following two concepts will play important roles in the coming arguments. Wedefine the oscillation ω M [s, t] of M over [s, t] by ω M [s, t] = sup s≤u,r≤t |M u −M r |. Withπ = (t 0 , . . . , t n ) a partition, by δ π (M) we denote the modulus of continuity over π,defined as δ π (M) = max 0≤i

286 Auxiliary resultsLemma C.1.3. Let π be a partition of [s, t]. It holds thatEQ π (M) 2 ≤ 3Eω 2 M [s, t]Q π (M).We also have the weaker bound EQ π (M) 2 ≤ 12‖M ∗ ‖ 2 ∞E(M t − M s ) 2 .Proof. We first expand the square to obtainEQ π (M) 2( n) 2∑= E (M tk − M tk−1 ) 2= Ek=1n∑(M tk − M tk−1 ) 4 + 2Ek=1n−1∑i=1 j=i+1The last term can be computed using Lemma 2.4.12,We then find,n−1∑2En−1∑= 2 Ei=1 j=i+1i=1⎛n∑(M ti − M ti−1 ) 2 (M tj − M tj−1 ) 2⎛⎝(M ti − M ti−1 ) 2 E ⎝n∑(M ti − M ti−1 ) 2 (M tj − M tj−1 ) 2 .∣ ⎞⎞n∑∣∣∣∣∣(M tj − M tj−1 ) 2 F ti⎠⎠j=i+1n−1∑= 2 E ( (M ti − M ti−1 ) 2 E((M t − M ti ) 2 |F ti ) )i=1n−1∑= 2 E(M ti − M ti−1 ) 2 (M t − M ti ) 2i=1n−1∑≤ 2EωM 2 [s, t] (M ti − M ti−1 ) 2i=1= 2Eω 2 M [s, t]Q π (M).EQ π (M) 2 ≤ En∑(M tk − M tk−1 ) 4 + 2EωM 2 [s, t]Q π (M)k=1≤ Eω 2 M [s, t]≤n∑(M tk − M tk−1 ) 2 + 2EωM 2 [s, t]Q π (M)k=13Eω 2 M [s, t]Q π (M),as desired. Since ω 2 M [s, t] ≤ 2‖M ∗ ‖ ∞ , we also getEQ π (M) 2 ≤ 12‖M ∗ ‖ 2 ∞EQ π (M) = 12‖M ∗ ‖ 2 ∞E(M t − M s ) 2 .

C.1 A proof for the existence of [M] 287Lemma C.1.4. Let π be a partition of [s, t]. Then(E sup (Mu − M s ) 2 − Q πu (M) ) 2≤ 17Eω2M [s, t] ( (M t − M s ) 2 + Q π (M) ) .s≤u≤tProof. Considering ((M u −M s ) 2 −Q π u(M)) 2 , we note that both terms in the square arenonnegative. Therefore, the mixed term in the expansion of the square is nonpositiveand we obtain the bound(E sup (Mu − M s ) 2 − Q π u(M) ) 2≤ E sup (M u − M s ) 4 + E sup Q π u(M) 2 .s≤u≤ts≤u≤ts≤u≤tThe proof of the lemma will proceed by first making an estimate that allows us to getrid of the suprema above. Then, we will evaluate the resulting expression to obtainthe result of the lemma. Specifically, we will remove the suprema by provingE sup (M u − M s ) 4 + E sup Q π u(M) 2s≤u≤ts≤u≤t≤ 5E(M t − M s ) 4 + 2Eω 2 M [s, t]Q π (M) + 5EQ π (M) 2 .This is a somewhat intricate task. We will split it into two parts, yielding the followingtotal structure of the proof: We first estimate the term E sup s≤u≤t (M u − M s ) 4 , thenestimate the term E sup s≤u≤t Q πu (M) 2 , and ultimately, we finalize our estimates andconclude.Step 1: Removing the suprema, first term. We begin by considering the termgiven by E sup s≤u≤t (M u − M s ) 4 . Note that whenever 0 ≤ u ≤ r, we have the equalityE(M s+r − M s |F s+u ) = M s+u − M s . Therefore, M s+u − M s is a martingale in u withrespect to the filtration (F s+u ) u≥0 . Since the dual exponent of 4 is 4 3, the Doob Lpinequality yields( ) 4 4E sup (M u − M s ) 4 ≤ E(M t − M s ) 4 ≤ 5E(M t − M s ) 4 ,s≤u≤t3removing the supremum.Step 2: Removing the suprema, second term. Turning our attention to the

288 Auxiliary resultssecond term, E sup s≤r≤t Q π r(M) 2 , let s ≤ u ≤ t. We know that==Q πu (M) 2(∑ n) 2(M tk ∧u − M tk−1 ∧u) 2k=1n∑(M tk ∧u − M tk−1 ∧u) 4 +k=1i≠jn∑(M ti∧u − M ti−1∧u) 2 (M tj∧u − M tj−1∧u) 2 .Therefore, we also have≤E sup Q π u(M) 2s≤u≤tn∑E sup (M tk ∧u − M tk−1 ∧u) 4s≤u≤tk=1+ E sups≤u≤tn∑(M ti ∧u − M ti−1 ∧u) 2 (M tj ∧u − M tj−1 ∧u) 2 .i≠jWe need to bound each of the two terms above. We begin by developing a bound forthe first term. Applying Doob’s L p inequality used with the martingale M t k− M t k−1yieldsE sups≤u≤tk=1n∑n∑(M tk ∧u − M tk−1 ∧u) 4 ≤ E sup (M t ku − M t k−1u ) 4s≤u≤tk=1n∑ 256≤81 E(M t kt − M t k−1t ) 4k=1n∑≤ 5 E(M tk − M tk−1 ) 4 .Next, we consider the second sum, that is,E sups≤u≤tk=1n∑(M ti ∧u − M ti−1 ∧u) 2 (M tj ∧u − M tj−1 ∧u) 2 .i≠jWe start out by fixing s ≤ u ≤ t. Assume that u < t, the case u = t will trivially satisfythe bound we are about to prove. Letting c be the index such that t c−1 ≤ u < t c , wecan separate the terms where one of the indicies is equal to c and obtain three typesof terms: Those where i is distinct from c and j is equal to c, those where i is equal toc and j is distinct from c and those where i and j are distinct and both distinct fromc as well. Note that since we are only considering off-diagonal terms, we do not need

C.1 A proof for the existence of [M] 289to consider the case where i and j are both equal to c. We obtain≤++n∑(M ti∧u − M ti−1∧u) 2 (M tj∧u − M tj−1∧u) 2i≠jn∑(M ti ∧u − M ti−1 ∧u) 2 (M u − M tc−1 ) 2i≠cn∑(M u − M tc−1 ) 2 (M tj ∧u − M tj−1 ∧u) 2j≠cn∑(M ti∧u − M ti−1∧u) 2 (M tj∧u − M tj−1∧u) 2 .i≠j,i≠c,j≠cNow note that when i < c, t i ∧ u = t i and t i−1 ∧ u = t i−1 . And when i > c, t i ∧ u = uand t i−1 ∧ u = u. Therefore,n∑(M ti ∧u − M ti−1 ∧u) 2 (M u − M tc−1 ) 2 =i≠cand analogously for the second term. Likewise,n∑(M ti − M ti−1 ) 2 (M u − M tc−1 ) 2i

290 Auxiliary resultsSince this bound holds for all s ≤ u ≤ t, we finally obtain≤E sups≤u≤tn∑(M ti ∧u − M ti−1 ∧u) 2 (M tj ∧u − M tj−1 ∧u) 2i≠j2Eω 2 M [s, t]Q π (M) + En∑(M ti − M ti−1 ) 2 (M tj − M tj−1 ) 2 .i≠jCombining our results, we finally obtainE sup Q πu (M) 2s≤u≤tn∑≤ 5 E(M tk − M tk−1 ) 4 + 2EωM 2 [s, t]Q π (M) + Ek=1≤ 2Eω 2 M [s, t]Q π (M) + 5EQ π (M) 2 .n∑(M ti − M ti−1 ) 2 (M tj − M tj−1 ) 2We have now developed the bounds announced earlier, and may all in all concludeas promised.i≠jE sup (M r − M s ) 4 + E sup Q π r(M) 2s≤r≤ts≤r≤t≤ 5E(M t − M s ) 4 + 2Eω 2 M [s, t]Q π (M) + 5EQ π (M) 2 ,Step 3: Final estimates and conclusion. We will now make an estimate for thefirst and last terms in the expression above, uncovering the result of the lemma. Forthe first term, we easily obtainFor the other term, Lemma C.1.3 yieldsCombining our findings, we conclude that5E(M t − M s ) 4 ≤ 5Eω 2 M [s, t](M t − M s ) 2 .5EQ π (M) 2 ≤ 15Eω 2 M [s, t]Q π (M).(E sup (Mu − M s ) 2 − Q πu (M) ) 2s≤u≤t≤ 5E(M t − M s ) 4 + 2Eω 2 M [s, t]Q π (M) + 5EQ π (M) 2≤5Eω 2 M [s, t](M t − M s ) 2 + 2Eω 2 M [s, t]Q π (M) + 15Eω 2 M [s, t]Q π (M)≤ 17Eω 2 M [s, t] ( (M t − M s ) 2 + Q π (M) ) ,which was precisely the proclaimed bound.

C.1 A proof for the existence of [M] 291Lemma C.1.5. Consider the space of processes on [0, t] endowed with the norm givenby ‖| · ‖| = ‖‖ · ‖ ∞ ‖ 2 . We denote convergence in ‖| · ‖| as uniform L 2 convergence.This space is complete.Proof. Assume that X n is a cauchy sequence under ‖| · ‖|. Since a cauchy sequencewith a convergent subsequence is convergent, it will suffice to show that X nk has aconvergent subsequence.We know that for any ε > 0, there exists N such that for n, m ≥ N, it holds that‖‖X n − X m ‖ ∞ ‖ 2 ≤ ε. In particular, using 1 as our ε, there exists n8 k k such that forany n, m ≥ n k ,P(‖X n − X m ‖ ∞ > 1 )2 k ≤ 4 k ‖‖X n − X m ‖ ∞ ‖ 2 ≤ 1 2 k .We can assume without loss of generality that n k is increasing. We then obtain thatP (‖X nk − X nk+1 ‖ ∞ > 1 ) ≤ 1 , so ‖X2 k 2 k nk − X nk+1 ‖ ∞ ≤ 1 from a point onwards2 kalmost surely, by the Borel-Cantelli Lemma. Then we also find for i > j‖X ni − X nj ‖ ∞ ≤j∑k=i+1‖X nk−1 − X nk ‖ ∞ ≤ 1 2 k ,so X nk is almost surely uniformly cauchy. Therefore, it is almost surely uniformlyconvergent. Let X be the limit. We wish to prove that X is the limit of X nk in ‖| · ‖|.Fixing k, we have‖|X nk − X‖| = ‖‖X nk − X‖ ∞ ‖ 2= ‖ lim ‖X nk − X nm ‖ ∞ ‖ 2m( () ) 12= E lim inf ‖X 2nm k− X nm ‖ ∞(= E lim inf (‖X nm k− X nm ‖ ∞ ) 2) 1 2= lim inf ‖‖X nm k− X nm ‖ ∞ ‖ 2 .Now note that X nk also is a cauchy sequence. Let ε > 0 be given and select N inaccordance with the cauchy property of X nk . Let k ≥ N, we then obtainlim infm‖‖X n k− X nm ‖ ∞ ‖ 2 = sup inf ‖‖X n k− X nm ‖ ∞ ‖ 2 ≤ ε.m≥ii≥NWe have now shown that for k ≥ N, ‖|X nk − X‖| ≤ ε. Thus, X nk converges to X,and therefore X n converges to X as well.

292 Auxiliary resultsWe are now ready to prove the convergence result needed to argue the existence ofthe quadratic variation. Letting π be a partition of [s, t], we earlier defined π u byπ u = {u} ∪ π ∩ [s, u] for s ≤ u ≤ t. We now extend this definition by defining π u beempty for u < s and letting π u = π for u > t. We use the convention that Q ∅ (M) = 0.Theorem C.1.6. Let t ≥ 0, and let π denote a generic partition of [0, t]. The mappings ↦→ Q πs (M) is uniformly convergent on [0, t] in L 2 .Proof. Since uniform L 2 convergence is a complete mode of convergence by LemmaC.1.5, it will suffice to show that s ↦→ Q π s(M) is a cauchy net with respect to uniformL 2 convergence. We therefore consider the distance between elements u ↦→ Q π u(M)and u ↦→ Q ϖu (M), where ϖ ⊇ π. As usual, π = (t 0 , . . . , t n ). We put ϖ = (s 0 , . . . , s m )and let j k denote the unique element of ϖ such that s jk = t k . We define the partitionϖ k of [t k−1 , t k ] by ϖ k = (s jk−1 , . . . , s jk −1) and obtainQ ϖu (M) − Q πu (M) ===m∑n∑(Ms u k− Ms u k−1) 2 − (Mt u k− Mt u k−1) 2k=1n∑j k∑k=1 i=j k−1 +1k=1n∑(Ms u i− Ms u i−1) 2 − (Mt u k− Mt u k−1) 2k=1n∑∑nQ ϖk u (M) − (Mt u k− Mt u k−1) 2 .k=1k=1This implies that=≤≤≤‖ sup |Q ϖ u(M) − Q π u(M)|‖ 2u≤t∥ supn∑n∣∥ ∑∣∣∣∣ ∥∥∥∥ Q ϖk u (M) − tu≤t ∣k=1k=1(M u k− Mt u k−1) 2 2∥ n∑∣ ∥∥∥∥∣sup ∣Q ϖk u (M) − (Mu∥tk− Mt u k−1) 2 ∣∣u≤tk=12n∑∣∥ ∥ sup ∣∣Q ϖk u (M) − (Mutk− Mt u k−1) 2 ∣∣ ∥∥∥u≤tk=12∥n∑∣ ∣ ∥∥∥∥2 ∣∣Q ϖ supk u (M) − (Mu∥tk− Mt u k−1) 2 ∣∣.t k−1 ≤u≤t kk=1We have been able to reduce the supremum to [t k−1 , t k ] since the expression in thesupremum is constant for u ≤ t k−1 and for u ≥ t k , respectively. Now, using Lemma

C.1 A proof for the existence of [M] 293C.1.4, we conclude≤≤∥n∑∣ ∣ ∥∥∥∥2∣∣Q ϖ supk u (M) − (Mu∥tk− Mt u k−1) 2 ∣∣t k−1 ≤u≤t kn∑17EωM 2 [t k−1 , t k ] ( (M tk − M tk−1 ) 2 + Q ϖ k(M) )k=1k=1n∑17EδM 2 [s, t] ( (M tk − M tk−1 ) 2 + Q ϖ k(M) ) .k=1Since the union of the ϖ k is ϖ, we obtain, applying the Cauchy-Schwartz inequalityand Lemma C.1.3,n∑17EδM 2 [s, t] ( (M tk − M tk−1 ) 2 + Q ϖ k(M) )k=1= 17Eδ 2 M [s, t](Q π (M) + Q ϖ (M))≤ 17‖δ 2 M [s, t]‖ 2 ‖Q π (M) + Q ϖ (M)‖ 2≤ 17‖δ 2 M [s, t]‖ 2 (‖Q π (M)‖ 2 + ‖Q ϖ (M)‖ 2 )≤ 34‖δM 2 ([s, t]‖ 2 12‖M ∗ ‖ 2 ∞E(M t − M s ) 2) 1 2=(34 √ )12‖M ∗ ‖ ∞ ‖M t − M s ‖ 2 ‖δM 2 [s, t]‖ 2 .As the mesh tends to zero, the continuity of M yields that δ π (M) tends to zero.By boundedness of M, Eδ π (M) 4 tends to zero as well. Therefore, if π is chosensufficiently fine, the above can be made as small as desired. Therefore, we concludethat s ↦→ Q π s(M) is uniformly L 2 cauchy, and so it is convergent.Theorem C.1.7. There exists a unique continuous, increasing and adapted process[M] such that for any t ≥ 0, [M] t is the uniform L 2 limit in π of the processess ↦→ Q πs (M) on [0, t].Proof. Letting n ≥ 1 and letting π be a generic partition of [0, n], by Lemma C.1.6,u ↦→ Q π u(M) is uniformly L 2 convergent. Let X n be the limit. X n is then a processon [0, n]. By Lemma C.1.2, there is a sequence of partitions π n such that u ↦→ Q πn u (M)converges uniformly in L 2 to X n . Therefore, there is a subsequence converging almostsurely uniformly. Since u ↦→ Q πn u (M) is continuous and adapted, X n is continuousand adapted as well.Now, X n is also the uniform L 2 limit of u ↦→ Q π u(M), u ≤ t, where π is a genericpartition of [0, n + 1]. By uniqueness of limits, X n and X n+1 are indistinguishable

294 Auxiliary resultson [0, n]. By the Pasting Lemma, there exists a process [M] such that [M] n = X n .Since X n is continuous and adapted, [M] is continuous and adapted as well. Since[M] t is the limit of larger partitions than [M] s whenver s ≤ t, [M] t ≥ [M] s and [M]is increasing. And [M] has the property stated in the lemma of being the uniform L 2limit on compacts of quadratic variations over partitions.Lemma C.1.8. The process M 2 t − [M] t is a uniformly integrable martingale.Proof. We proceed in two steps. We first show that M 2 t − [M] t is a martingale, andthen proceed to show that it is uniformly integrable.Step 1: The martingale property. Let 0 ≤ s ≤ t. Let π be a partition of [0, t].Define ϖ = {s} ∪ π ∩ [s, t]. We then obtainThis shows thatE(Q π (M)|F s ) = E(Q π s(M) + Q ϖ (M)|F s )= Q π s(M) + E((M t − M s ) 2 |F s )= Q π s(M) + E(M 2 t |F s ) − M 2 s .E(M 2 t − Q π t(M)|F s ) = E(M 2 t − Q π (M)|F s ) = M 2 s − Q π s(M),so the process M 2 s −Q πs (M) is a martingale on [s, t]. Since Q πs (M) converges in L 2 to[M] s as π becomes finer, and conditional expectations are L 2 continuous, we concludeE(M 2 t − [M] t |F s ) = limπE(M 2 t − Q π t(M)|F s )= limπM 2 s − Q π s(M)= M 2 s − [M] s ,almost surely, showing the martingale property.Step 2: Uniform integrability. By the continuity of the L 2 norm, we can useLemma C.1.3 to obtainE[M] 2 t = limπE(Q π (M) 2 ) ≤ 12‖M ∗ ‖ 2 ∞EM 2 t ≤ 12‖M ∗ ‖ 4 ∞,so [M] is bounded in L 2 , therefore uniformly integrable. Since M 2 is bounded, it isclearly uniformly integrable. We can therefore conclude that M 2 t − [M] t is a uniformlyintegrable martingale.

C.1 A proof for the existence of [M] 295Our final task is localise our results from continuous bounded martingales to continuouslocal martingales and investigate how to characterize the quadratic variation in thelocal cases.Lemma C.1.9. Let π be a partition of [s, t]. Let τ be a stopping time with values in[s, ∞). ThenQ π τ(M) = Q π (M τ ).Proof. This follows immediately fromQ πτ (M) ==n∑(Mt 2 k ∧τ − Mt 2 k−1 ∧τ )k=1n∑((Mt τ k) 2 − (Mt τ k−1) 2 )k=1= Q π (M τ ).Lemma C.1.10. Let M be a continuous bounded martingale, and let τ be a stoppingtime. Then [M] τ = [M τ ].Proof. Let t ≥ 0. As usual, letting π be a generic partition of [0, t], we know that [M]is the uniform L 2 limit on [0, t] of Q π s(M).E ([M] τ t − Q π (M τ )) 2 = E ([M] τ t − Q π (M) τ ) 2= E ([M] τ t − Q π τ(M)) 2(2≤ E sup |[M] s − Q (M)|) πs ,s≤twhich tends to zero. Thus, [M] τ t is the limit of Q π (M τ ). Since [M τ ] t is also thelimit of Q π (M τ ), we must have [M] τ t = [M τ ] t . Since the processes are continuous, weconclude that [M] τ = [M τ ] up to indistinguishability.Corollary C.1.11. Let M be a continous local martingale. There exists a uniquecontinuous, increasing and adapted process [M] such that if τ n is any localising sequencesuch that M τ nis a continuous bounded martingale, then [M] τ n= [M τ n].Proof. This follows from Theorem 2.5.11.

296 Auxiliary resultsLemma C.1.12. Let A be any increasing process with A 0 = 0. Then A is uniformlyintegrable if and only if A ∞ ∈ L 1 .Proof. First note that the criterion in the lemma is well-defined, since A is convergentand therefore has a well-defined limit in [0, ∞]. We show each implication separately.Assume A ∞ ∈ L 1 . We then obtain, since A is increasing,lim sup E|A t |1 (|At|>x) ≤ limx→∞t≥0supx→∞ t≥0E|A ∞ |1 (|A∞|>x) = 0,by dominated convergence. Therefore, A is uniformly integrable. On the other hand,if A is uniformly integrable, we find by monotone convergenceE|A ∞ | = limtE|A t | ≤ sup E|A t |,t≥0which is finite, since A is uniformly integrable and therefore bounded in L 1 .Theorem C.1.13. Let M ∈ cM 2 0. The quadratic variation is the unique process [M]such that M 2 − [M] is a uniformly integrable martingale.Proof. Let τ n be a localising sequence for M such that M τn is a continuous boundedmartingale. We will use Lemma 2.4.13 to prove the claim. Therefore, we first provethat M 2 −[M] has an almost surely finite limit. We already know that M 2 has a limit,it will therefore suffice to show that this is the case for [M]. This follows sinceE[M] ∞ = E limn[M] τn= limnE[M] τn= limnE[M τn ] ∞= limnE(M τn ) 2 ∞≤ lim EM 2 ∞,which is finite. Now let a stopping time τ be given. Using that M ∈ cM 2 0 and applyingmonotone convergence, we obtainE(Mτ 2 − [M] τ ) = EMτ 2 − E[M] τ= E lim Mτ 2 n n∧τ − E lim[M] τn ∧τn= lim EMτ 2 n n∧τ − lim E[M] τn ∧τn= − lim E(M τn ) 2 τ − [M τn ] τn= 0.

C.2 A Lipschitz version of Urysohn’s Lemma 297By Lemma 2.4.13, M 2 − [M] is a uniformly integrable martingale.Corollary C.1.14. Let M ∈ cM L 0 . The quadratic variation is the unique increasingprocess [M] such that M 2 − [M] is a continuous local martingale.C.2 A Lipschitz version of Urysohn’s LemmaIn this section, we prove an version of Urysohn’s Lemma giving a Lipschitz conditionon the mapping supplied. This result could be used in the proof of Lemma 4.3.3, butthe proof presented there seems simpler. However, since the extension of Urysohn’sLemma has independent interest, we present it here.Lemma C.2.1. Let ‖ · ‖ be any norm on R n and let d be the induced metric. Let Abe a closed set and let x ∈ A c . Then d(x, A) = d(x, y) for some y ∈ ∂A.Proof. First consider the case where A is bounded. Then A is compact. The mappingy ↦→ d(x, y) is continuous, and therefore attains its infimum over A. Thus, there existsy ∈ A such that d(x, A) = d(x, y). Assume that y ∈ A ◦ , we will seek a contradiction.Let ε > 0 be such that B ε (y) ⊆ A. There exists 0 < λ ≤ 1 with λx + (1 − λ)y ∈ B ε (y).We then find‖λx + (1 − λ)y − x‖ ≤ (1 − λ)‖x − y‖ < ‖x − y‖ = d(x, y),and therefore d(x, λx + (1 − λy)) < d(x, y), which is the desired contradiction. Thus,y /∈ A ◦ , and since A is closed, we obtain y ∈ ∂A.Next, consider the case where A is unbounded. Then, there exists a sequence z n in Awith norms converging to infinity. We defineK n = {y ∈ A|‖y − x‖ ≤ ‖z n − x‖}.Since the norm is continuous, K n is closed. And any y ∈ K n satisfies the relation‖y‖ ≤ ‖y − x‖ + ‖x‖ ≤ ‖z − x‖ + ‖x‖, so K n is bounded. For any y ∈ A \ K n , we haved(y, x) > d(x, z n ) ≥ d(x, A) and therefore d(x, A) = d(x, K n ).From what we already have shown, there is y n ∈ ∂K n such that d(x, A) = d(x, y n ).We need to prove that y n ∈ ∂A for some n. Note that with α n = ‖z n − x‖, we have∂K n = ∂(A ∩ B αn (x)) ⊆ ∂A ∪ ∂B αn (x) = ∂A ∪ {y ∈ R n |‖y − x‖ = α n }.

298 Auxiliary resultsSince ‖y n − x‖ = d(y n , x) = d(x, A), the sequence ‖y n − x‖ is constant, in particularbounded. But α n tends to infinity, since ‖z n ‖ tends to infinity. Thus, there is n suchthat ‖y n − x‖ ̸= α n , and therefore y n ∈ ∂A, as desired.Theorem C.2.2 (Urysohn’s Lemma with Lipschitz constant). Let K and Vbe compact, respectively open subsets of R n , with K ⊆ V . Let ‖ · ‖ be any norm onR n , and let d be the induced metric. Assume that V has compact closure. There existsf ∈ C c (R n ) such that K ≺ f ≺ V and such that f is d-Lipschitz continuous andd-Lipschitz constant4 sup x∈∂V,y∈∂K d(x, y)(inf x∈V \K ◦ d(x, K) + d(x, V c )) 2 .Proof. Clearly, we can assume K ≠ ∅. We definef(x) =d(x, V c )d(x, K) + d(x, V c ) ,and claim that f satisfies the properties given in the lemma. We first argue that f iswell-defined. We need to show that the denominator in the expression for f is finiteand nonzero. Since K ≠ ∅, d(x, K) is always finite. Since V has compact closure,V ≠ R n , so d(x, V c ) is also always finite. Thus, the denominator is finite. It is alsononnegative, and if d(x, K) + d(x, V c ) = 0, we would have by the closedness of K andV c that x ∈ K ∩ V c , an impossibility. This shows that f is well-defined.It remains to prove that f ∈ C c (R n ), that K ≺ f ≺ V and that f has the desiredLipschitz properties. We will prove these claims in four steps.Step 1: f ∈ C c (R n ) with K ≺ f ≺ V . It is clear that f is continuous. If x ∈ K,d(x, K) = 0 and therefore f(x) = 1. If x ∈ V c , d(x, V c ) = 0 and f(x) = 0. Thus,K ≺ f ≺ V . Also, since f is zero outside of V , the support of f is included in V .Since V has compact closure, f has compact support.Step 2: Lipschitz continuity of f on V \ K ◦ . We begin by showing that f isLipschitz continuous on the compact set V \ K ◦ . Letting x, y ∈ V \ K ◦ , we findf(x) − f(y) =d(x, V c )d(x, K) + d(x, V c ) − d(y, V c )d(y, K) + d(y, V c )= d(x, V c )(d(y, K) + d(y, V c )) − d(y, V c )(d(x, K) + d(x, V c ))(d(x, K) + d(x, V c ))(d(y, K) + d(y, V c ))d(x, V c )d(y, K) − d(y, V c )d(x, K)=(d(x, K) + d(x, V c ))(d(y, K) + d(y, V c )) .

C.2 A Lipschitz version of Urysohn’s Lemma 299Since the mapping x ↦→ d(x, K) + d(x, V c ) is positive and continuous in x and y andV \ K ◦ is compact, we conclude that the infimum of d(x, K) + d(x, V c ) over x inV \ K ◦ is positive. Let C denote this infimum. Furthermore, note that the mappingsx ↦→ d(x, V c ) and x ↦→ d(x, K) is continuous, and therefore their suprema over x inV \ K ◦ are finite. Let c 1 and c 2 denote these suprema. Note that for any x, y, a, b ∈ R,( a + bxa − yb = x2= (x − y) a + b2+ a − b2)− y( a + b2+ (a − b) x + y ,2− a − b2so by symmetry, |xa − yb| ≤ |x − y| |a+b|2+ |a − b| |x+y|2. This shows that|d(x, V c )d(y, K) − d(y, V c )d(x, K)||f(x) − f(y)| =(d(x, K) + d(x, V c ))(d(y, K) + d(y, V c ))≤ 1C 2 |d(x, V c )d(y, K) − d(y, V c )d(x, K)|≤ 1C 2 (c 2|d(x, V c ) − d(y, V c )| + c 1 |d(x, K) − d(y, K)|)≤ 1C 2 (c 2d(x, y) + c 1 d(x, y))≤ c 1 + c 2C 2 d(x, y).Now, since , We have now shown the Lipschitz property of f for x, y ∈ V , obtainingthe Lipschitz constant c = c1+c2C. We will now extend the Lipschitz property to all of2R n .)Step 3: Lipschitz property of f on R n . We first extend the Lipschitz propertyto R n \ K ◦ . Let x, y ∈ R n \ K ◦ be arbitrary. Since we already have the Lipschitzproperty when x, y ∈ V \ K ◦ , we only need to consider the case where one of x andy are in V c . If x and y both are in V c , they are also both in V c , so f(x) − f(y) = 0and the Lipschitz property trivially holds here. It remains to consider the case where,say, x ∈ V c and y ∈ V \ K ◦ . Now, since V is compact, by Lemma C.2.1 there existsz ∈ ∂V such that d(x, V ) = d(x, z). Now, since V is open, ∂V = ∂V = ∂V c ⊆ V c .Since f is zero on V c , we conclude f(z) = 0. Since also f(x) = 0, we obtain|f(x) − f(y)| ≤ |f(x) − f(z)| + |f(z) − f(y)| = |f(y) − f(z)| ≤ cd(y, z).Now, z is the element of V which minimizes the distance to x. Since y ∈ V , it followsthat d(x, z) ≤ d(x, y) and therefore d(y, z) ≤ d(y, x) + d(x, z) ≤ 2d(x, y). We conclude|f(x) − f(y)| ≤ 2cd(y, z).

300 Auxiliary resultsWe have now proven that f is Lipschitz on R n \ K ◦ with Lipschitz constant 2c.Finally, we extend the Lipschitz property to all of R n . Let x, y ∈ R n . As before, itwill suffice to consider the case where either x ∈ K ◦ or y ∈ K ◦ . In the case x, y ∈ K ◦ ,f(x) − f(y) = 0 and the Lipschitz property is trivial.We therefore consider the case where x ∈ K ◦ and y ∈ R n \K ◦ . By Lemma C.2.1, thereis z on the boundary of R n \ K ◦ such that d(x, V \ K ◦ ) = d(x, z). Now, the boundaryof R n \ K ◦ is the same as ∂K, so z ∈ ∂K. K is closed and therefore ∂K ⊆ K, sof(z) = 1. Since also x ∈ K ◦ ⊆ K, f(x) = 1 and we find|f(x) − f(y)| ≤ |f(x) − f(z)| + |f(z) − f(y)| = |f(y) − f(z)|.We know that f is Lipschitz on R n \ K ◦ and that y ∈ R n \ K ◦ . Since z ∈ ∂(R n \ K ◦ ),there is a sequence z n ∈ R n \ K ◦ such that z n converges to z. This yields|f(y) − f(z)| = limn|f(y) − f(z n )| ≤ limncd(y, z n ) = cd(y, z),and we thus recover |f(x) − f(y)| ≤ cd(y, z). As in the previous step, d(x, z) ≤ d(x, y)and so d(y, z) ≤ d(y, x) + d(x, z) ≤ 2d(x, y), yielding |f(x) − f(y)| ≤ 2cd(y, z). Thisshows the Lipschitz property.Step 4: Estimating the Lipschitz constant. We now know that f is Lipschitzcontinuous with Lipschitz constant2c = 2c 1 + 2c 2C 2 = 2 sup x∈V \K ◦ d(x, K) + 2 sup x∈V \K ◦ d(x, V c )(inf x∈V \K ◦ d(x, K) + d(x, V c )) 2 .This expression is somewhat cumbersome. By estimaing it upwards, we obtain aweaker result, but more manageable. We will leave the denominator untouched andonly consider the numerator. Our goal is to showsup d(x, K) ≤ sup d(x, y)x∈V \K ◦ x∈∂V,y∈∂Ksup d(x, V c ) ≤ sup d(x, y).x∈V \K ◦ x∈∂V,y∈∂KConsider the first equation. Obviously, to prove the inequality, we can assume thatsup x∈V \K ◦ d(x, K) is nonzero. First note that since the mapping x ↦→ d(x, V c ) iscontinuous and V \ K ◦ is compact, there is z ∈ V \ K ◦ such thatsup d(x, V c ) = d(z, V c ).x∈V \K ◦

C.2 A Lipschitz version of Urysohn’s Lemma 301And by Lemma C.2.1, there is w ∈ ∂V c = ∂V such that d(z, V c ) = d(z, w). Wewant to prove z ∈ ∂K. First assume that z is in the interior of V \ K ◦ , we seek acontradiction. Let ε > 0 be such that B ε (z) ⊆ V \ K ◦ . There is 0 < λ ≤ 1 such thatλw + (1 − λ)z ∈ B ε (z), and we then findd(λw + (1 − λ)z, w) = ‖(1 − λ)(z − w)‖ = (1 − λ)‖z − w‖ < d(z, w).But z was the infimum of d(y, w) over y ∈ V \ K ◦ . Thus, we have obtained a contradictionand conclude that z is not in the interior of V \ K ◦ . Since V \ K ◦ is closed, weconclude z ∈ ∂(V \ K ◦ ) ⊆ ∂V ∪ ∂K. If z ∈ ∂V , we obtain d(z, V c ) = 0, inconsistentwith our assumption that sup x∈V \K ◦ d(x, K) is nonzero. Therefore, z ∈ ∂K. We maynow concludesup d(x, K) = d(z, w) ≤ sup d(x, y).x∈V \K ◦ x∈∂V,y∈∂KWe can use exacly the same arguments to prove the other inequality. We can thereforeall in all conclude that f is Lipschitz with Lipschitz constantas desired.4 sup x∈∂V,y∈∂K d(x, y)(inf x∈V \K ◦ d(x, K) + d(x, V c )) 2 ,Comment C.2.3 The Lipschitz constant could possibly be improved by instead consideringthe mapping{ d(x, V c })f(x) = mind(K, V c ) , 1 .◦

302 Auxiliary results

Appendix DList of SymbolsGeneral probability• (Ω, F, F T , P ) - the background probability space.• F ∞ - The σ-algebra induced by ∪ t≥0 F t .• W - Generic Brownian motion.• τ, σ - Generic stopping times.• Σ A - The σ-algebra of measurability and adaptedness.• Σ π - The progressive σ-algebra.• M - Generic martingale.• M ∗ t- The maximum of M over [0, t].• M ∗ - The maximum of M over [0, ∞).• SP - The space of real stochastic processes.• M - The space of martingales.• M 0 - The space of martingales starting at zero.• cM 2 0 - The space of continuous square-integrable martingales starting at zero.

304 List of Symbols• cM L 0 - The space of continuous local martingales starting at zero.• P−→ - Convergence in probability.• −→ Lp- Convergence in L p .• a.s.−→ - Almost sure convergence.• φ - The standard normal density.• φ n - The n-dimensional standard normal density.Stochastic integration• bE - The space of elementary processes.• L 2 (W ) - The space of progressive processes X with E ∫ ∞0X 2 t dt finite.• L 2 (W ) - The space of processes L 0 -locally in L 2 (W ).• cM L W - The processes ∑ nk=1∫ t0 Y k dW k t , Y ∈ L 2 (W ) n .• µ M - The measure induced by M ∈ cM L W .• L 2 (M) - The L 2 space induced by µ M .• L 2 (M) - The space of processes L 0 -locally in L 2 (M).• cFV - The space of continuous finite variation processes.• cFV 0 - The space of continuous finite variation processes starting at zero.• S - The space of standard processes.• L(X) - The space of integrands for X ∈ S.• [X] - The quadratic variation of a process X ∈ S.• Q π (M) - The quadratic variation of M over the partition π.• E(X) - The Doléans-Dade exponential of X ∈ S.• X, Y - Generic elements of S.• H, K - Generic elements of bE.

305The Malliavin calculus• D - The Malliavin derivative.• S - The space of smooth variables, Cp ∞ (R n ) transformations of W coordinates.• L p (Π) - The space L p ([0, T ] × Ω, B[0, T ] ⊗ F T , λ ⊗ P ).• D 1,p - The domain of the extended Malliavin derivative.• F - Generic element of D 1,p .• X - Generic element of L 2 (Π).• θ - The Brownian stochastic integral operator on [0, T ].• (e n ) - Generic countable orthonormal basis of L 2 [0, T ].• 〈·, ·〉 [0,T ] - The inner product on L 2 [0, T ].• 〈·, ·〉 Π - The inner product on L 2 (Π).• H n - The subspace of L 2 (F T ) based on the n’th Hermite polynomial.• H n,H ′ n ′′ - Dense subspaces of H n .• H n - Orthonormal basis of H n .• H n (Π) - The subspace of L 2 (Π) based on H n .• H n(Π), ′ H n(Π) ′′ - Dense subspaces of H n (Π).• H n (Π) - Orthonormal basis for H n (Π).• P n - Orthogonal projection onto H n .• P Π n- Orthogonal projection onto H n (Π).• Φ a - Generic element of H n .• P n - Polynomials of degree n in any number of variables.• P n - Subspace of L 2 (F T ) based on P n .• P n ′ - Dense subspace of P n .• P n - Total subset of P n .

306 List of Symbols• δ - The Skorohod integral.Mathematical finance• Q - A generic T -EMM.• S - Generic asset price process.• r - Generic short rate process.• B - Generic risk-free asset price process.• S ′ - Generic normalized asset price process.• B ′ - Generic normalized risk-free asset price process.• M - Generic financial market model.• (h 0 , h S ) - Generic portfolio strategy.• V h - Value process of the portfolio strategy h.• µ - Generic drift vector process.• σ - Generic volatility matrix process.• ρ - Generic correlation.Measure theory• B - The Borel σ-algebra on R.• B k - The Borel σ-algebra on R k .• B(M) - The Borel σ-algebra on a pseudoemetric space (M, d).• λ - The Lebesgue measure.• [f] - The equivalence class corresponding to f.• C([0, ∞), R n ) - The space of continuous mappings from [0, ∞) to R n .• Xt ◦ - The t’th projection on C([0, ∞), R n ).

307• C([0, ∞), R n ) - The σ-algebra on C([0, ∞), R n ) induced by the projections.Analysis• C 1 (R n ) - The space of continuous differentiably mappings from R n to R.• C ∞ (R n ) - The infinitely differentiable elements of C 1 (R n ).• C ∞ p (R n ) - The elements of C ∞ (R n ) with polynomial growth for the mappingand its partial derivatives.• C ∞ c(R n ) - The elements of C ∞ (R n ) with compact support.• C 2 (U) - The space of twice continuously differentiable mappings from U to R n .• ‖ · ‖ ∞ - The uniform norm.• H n - The n’th Hermite polynomial.• M ⊕ N - orthonormal sum of two closed subspaces.• V F - Variation of F .• µ F - Signed measure induced by a finite variation mapping F .• f ∗ g - Convolution of f and g.• (ψ ε ) - Dirac family.• K - Generic compact set.• U, V - Generic open sets.

308 List of Symbols

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