On Dufresne's translated perpetuity and some Black-Scholes annuities

This is somewhat different from the price of a classic European Call option, where only the final value ofthe underlying asset at time T is considered (see [Cha12]) :[E (S T − K) +] .Of course, Asian options are harder to compute in practice as they depend on the entire past history of theunderlying asset, but they make it possible to reduce the risk of price manipulation near the maturity date.The paper is finally concluded by a short appendix on Bessel functions and Bessel processes (with drift).2 A generalization of Dufresne’s translated perpetuityIn this section, we compute the law of the perpetuities :∫ +∞0Es2 ∫ +∞(Es 2 + 2aE s + b) 2 ds and M 2 s(M 2 s + 2aM s + b) 2 ds.0Theorem 2. Assume that the polynomial z 2 + 2az + b does not have positive roots. For 2λ + a 2 − b ≥ 0,we have:( ∫ √2λ x)dz( ∫ +∞EsE x[exp2 )] √ sinh + a2 − b−λ0 (Es 2 + 2aE s + b) 2 ds x2 + 2ax + b0 z=2 + 2az + b( ∫x√2λ +∞)dzsinh + a2 − bz 2 + 2az + band( ∫ +∞ME x[exp2 )] √ sinhsb + 2ax + x−λ0 (M 2 s + 2aM s + b) 2 ds 2= √b( ∫ √2λ b/x+ a2 − bsinh( √2λ+ a2 − b00∫ +∞0)dzz 2 + 2az + b).dzz 2 + 2az + bThe equality in law given in Theorem 1 follows directly from this result, since when a 2 − b ≥ 0, onerecognizes in the right-hand side the expression of the Laplace transform of the first passage time of athree-dimensional Bessel process with drift √ a 2 − b. A short review of Bessel processes with drift is givenin Section 5, where these Laplace transforms are also inverted thanks to Jacobi’s theta function.2.1 A martingale approach to Dufresne’s translated perpetuityLet x > 0 and assume that E 0 = x. By Lamperti’s relation (see Theorem 7), there exists a threedimensionalBessel process (R t , t ≥ 0) started from x such that:E t = R At with A t = 〈E〉 t =Using this transform and the change of variable u = A s , we obtain :∫ +∞0Es2 ∫ +∞(Es 2 + 2aE s + b) 2 ds =0∫ t0(E s ) 2 ds.R 2 A s(R 2 A s+ 2aR As + b) 2 ds = ∫ +∞01(Ru 2 du. (2)+ 2aR u + b)2To compute the law of this perpetuity, we shall construct the appropriate martingale, thanks to thefollowing lemma:4

Lemma 3 ([PZ03]). Let ∆ = a 2 − b and 2λ + ∆ > 0. Set for x ≥ 0:Then, the functionsη(x) =are two independent solutions of the equation:∫ x0dzz 2 + 2az + b .φ ± (x) = √ (x 2 + 2ax + b exp ±η(x) √ )2λ + ∆f ′′ −2λ(x 2 + 2ax + b) 2 f = 0.Let (W t , t ≥ 0) be a Brownian motion. Applying Itô’s formula, we deduce that the process√(M t = Wt 2 + 2aW t + b sinh η(W t ) √ () ∫ )tdu2λ + ∆ exp −λ(Wu 2 + 2aW u + b) 2is a continuous martingale, and with T 0 = inf{t ≥ 0, W t = 0}, Doob’s optional stopping theorem implies:√x2 + 2ax + b sinh( √2λ+ ∆η(x))[ √ (= E x Wt∧T 2 0+ 2aW t∧T0 + b sinh η(W t∧T0 ) √ () ∫ t∧T02λ + ∆ exp −λ[ √ (= E x Wt 2 + 2aW t + b sinh η(W t ) √ () ∫ t2λ + ∆ exp −λ000)]du(Wu 2 + 2aW u + b) 2)du(Wu 2 + 2aW u + b) 2 1 {t

after the change of variable y = xz.=√b + 2ax + x2√b( ∫ )√2λb/xdysinh + ∆0 y 2 + 2ay + b( ∫ √2λ +∞).dysinh + ∆y 2 + 2ay + b0Remark 4. Letting x tend towards 0 in (3), we obtain the simple formula:[ ( ∫ )]+∞√E (3)du2λ + ∆0 exp −λ(Ru 2 + 2aR u + b) 2 = √ ( √ ).b sinh η(+∞) 2λ + ∆02.2 A probabilistic approach to Dufresne’s translated perpetuityWe now give another proof of Theorem 2, with a slightly more probabilistic approach. Let x > 0 andassume that M 0 = x. From the Dambis, Dubins-Schwarz theorem (see [RY99, Theorem 1.6, p.181]), thereexists a Brownian motion (B t , t ≥ 0) started from x such that:Therefore:∫ +∞0M 2 ∫ +∞s(M 2 s + 2aM s + b) 2 ds =M t = B 〈M〉t with 〈M〉 t =0∫ t0(M s ) 2 ds.B 2 〈M〉 s(B 2 〈M〉 s+ 2aB 〈M〉s + b) 2 ds = ∫ T001(Bu 2 du, (5)+ 2aB u + b)2where T 0 = inf{u ≥ 0, B u = 0}. By the classic time-reversal result of Brownian motion (see Section 5),this last expression is seen to be identical in law with:∫ T001(B 2 u + 2aB u + b) 2 du = ∫ T00∫1Gx(law)(BT 2 + 2aB ds =0−s T 0−s + b)201(Rs 2 ds (6)+ 2aR s + b)2where (R s , s ≥ 0) is a three-dimensional Bessel process starting from 0 and G x = sup{s ≥ 0, R s = x}.To compute the law of this last expression, we shall first consider the whole perpetuity∫ +∞0ds(R 2 s + 2aR s + b) 2 .From the occupation time formula and Ray-Knight’s third theorem (see Theorem 10):∫ +∞0∫ds+∞(Rs 2 + 2aR s + b) 2 =0L y ∞(R)(y 2 + 2ay + b)(law)2dy =∫ +∞0Z (2)y(y 2 + 2ay + b) 2 dywhere (Z y(2) , y ≥ 0) denotes a two-dimensional squared Bessel process starting from 0. Then, from Theorem8, the Laplace transform of the right-hand side equals :[ ( ∫ )]+∞Z y(2)E exp −λ(y 2 + ay + b) 2 dy = F (∞)where F is the unique solution on [0, +∞[ of :F ′′ =02λ(x 2 F such that F is positive, non increasing, and F (0) = 1.+ 2ax + b)26

Therefore, from Lemma 3, there exist two constants α and β such that :F (x) = √ ( (x 2 + 2ax + b α cosh η(x) √ ) (2λ + ∆ + β sinh η(x) √ ))2λ + ∆ .Since F (0) = 1, we deduce that:α = 1 √b.Next, as F is positive and non increasing, the limit F (∞) necessarily exists, so we must have :which yields1(√ cosh η(+∞) √ ) (2λ + ∆ + β sinh η(+∞) √ )2λ + ∆ = 0bβ = − 1 √bcosh ( η(+∞) √ 2λ + ∆ )sinh ( η(+∞) √ 2λ + ∆ ) .Finally, thanks to the additivity formula of sinh:√x2 + 2ax + b sinh (√ 2λ + ∆(η(∞) − η(x)) )√2λ + ∆F (x) = √b sinh ( η(+∞) √ 2λ + ∆ ) −−−−−→ √ ( √ )x→+∞ b sinh η(+∞) 2λ + ∆and we recover the result of Remark 4. Now, to obtain the result for any x > 0 we shall use a decompositionof the paths of the three-dimensional Bessel process at its last passage time G x = sup{t ≥ 0, R t = x}.From Theorem 9, we may write :∫ +∞0∫dsGx(Rs 2 + 2aR s + b) 2 ==(law)=0∫ Gx0∫ Gx0∫ds+∞(Rs 2 + 2aR s + b) 2 +G x∫ds+∞(Rs 2 + 2aR s + b) 2 +∫ds+∞(Rs 2 + 2aR s + b) 2 +00ds(R 2 s + 2aR s + b) 2ds(R2Gx+s + 2aR G x+s + b ) 2ds((x + ˜R s ) 2 + 2a(x + ˜R s ) + b(where ˜Rs , s ≥ 0)is an independent copy of (R s , s ≥ 0). Taking the Laplace transform of both sides, weobtain, from (5) and (6):[ (E (3)0= E (3)0exp[exp−λ(∫ +∞0−λ∫ Gx( ∫ +∞= E x[exp −λ0which yields the formula:0)]ds(Rs 2 + 2aR s + b) 2)]ds(Rs 2 + 2aR s + b) 2E (3)0⎡⎛⎢ ⎜⎣exp ⎝−λ∫ +∞M 2 )] [ ( ∫ +∞s(M 2 s + 2aM s + b) 2 ds E (3)0 exp −λ0( ∫ +∞ME x[exp2 )]s−λ0 (M 2 s + 2aM s + b) 2 ds0) 2⎞⎤⎟⎥) 2 ⎠⎦ds((x + ˜R s ) 2 + 2a(x + ˜R s ) + b)]ds(Rs 2 + (2a + 2x)R s + x 2 + 2ax + b) 2 ,7

( ∫ √2λ +∞)dz√ sinh + a2 − bx2 + 2ax + b0 z= √ 2 + 2(a + x)z + x 2 + 2ax + b( ∫ b√2λ +∞) .dzsinh + a2 − bz 2 + 2az + bThis new expression is seen to agree with Theorem 2 by applying the change of variable z = b y− x in theintegral on the numerator. The other relation follows as before thanks to (4).02.3 A few particular casesi) When a = 1 and b = 1, we recover a particular case of Hariya’s identity :∫ +∞with E 0 = x and R 0 = 1 − 11+x .00Es2 ∫ +∞(Es 2 + 2E s + 1) 2 ds = ds (law)(R s + 1) 4 = inf {t ≥ 0, R t = 1} .ii) More generally, when a = √ b,∫ +∞Es2 ∫ +∞{(Es 2 + 2bE s + b 2 ) 2 ds = ds (law)(R s + b) 4 = inf t ≥ 0, R t = 1 },bwith E 0 = x and R 0 = 1 b − 1x+b .00iii) We may recover the result of Salminen-Yor [SY05] by letting b → 0. Indeed, for 0 < b < a 2 , we have:∫ ( (xdzz 2 + 2az + b = 12 √ x + a − √ ) (aln2 − ba 2 − b x + a + √ a − √ ))a− ln2 − ba 2 − b a + √ a 2 − b0so that, letting b go towards 0, we obtain:( √2λ ∫ xsinh + a2 − b0and)dzz 2 + 2az + b∼b→0⎛ ⎛√ ( )12 exp ⎝ 2λ + a2 x⎝ln2a x + 2a( ∫ +∞)] √ ( )dsE x[exp√ 2λ+a 2x + 2a x2a−λ0 (E s + 2a) 2 = √ x x + 2a=⎛ √ ⎞⎞⎞− ln ⎝ 1 − 1 − ba 2⎠⎠⎠2( ) √ 2λ+a 2x2a − 1 2.x + 2aThis last expression is seen to coincide with the Laplace transform of the first hitting time at level12a ln ( )x+2ax of a Brownian motion with drift a started from 0, which was the announced result inthe introduction, with x = 1.We refer to Salminen & Yor [SY04, SY05] and Decamps, De Schepper, Goovaerts & Schoutens [DDSGS05]for similar articles on this subject.3 Some Black-Scholes annuitiesLet (B t + νt, t ≥ 0) be a standard Brownian motion with drift ν started from 0. In this section, we studythe law of the pair of annuities :(∫ t () +∫ t () ) −e 2β(Bs+νs) − 1 ds, e 2b(Bs+νs) − 1 ds (7)where x + = max(0, x) and x − = min(0, x).008

Theorem 5. Let α, β, a, b, λ ≥ 0 and ν ∈ R. The double Laplace transform of the couple (7) is given by:∫ +∞0(e −(λ+a)t E 0[exp −α( (√= 22αK 2γω λ βwhere the Wronskien ω λ equals:∫ t0) ∫ 0ω λ = √ 2aK 2γ(√2αβ() +∫ te 2β(Bs+νs) − 1 ds − a(√ ) (√2a2ae νy I 2c−∞ b eby dy + I 2cb)I ′ 2c0() )] −e 2b(Bs+νs) − 1 ds dt) ∫ +∞(√ )2a− √ (√ ) (√ )2α 2a2αK 2γ′ I 2cbβ band I 2c and K 2γ denote the third modified Bessel functions with respective indexes0e νy K 2γ(√2αβ eβy )dy⎧1 √2(λ + a) − 2α + ν2if 2(λ + a) − 2α + νc =2b√ 1 ⎪⎨2 ≥ 02β2λ + ν2and γ =⎪⎩i √2α − 2(λ + a) − ν2otherwise.2βThe proof of this result is given in the next Sections 3.1 and 3.2. We will discuss in Section 4 some specialcases for which the expression on the right-hand side simplifies.3.1 A useful version of the Feynman-Kac formulaTo prove Theorem 5, we shall apply the following well-known Feynman-Kac formula, see for instanceJanson [Jan07, Appendix C] where many other Brownian areas are also studied.Theorem 6 (Feynman-Kac). Let V (x) ≥ 0 be a positive continuous function on R, λ > 0, and let φ +and φ − be two C 2 -solutions of the differential equation)such that, for A large enough :12 φ′′ (x) = (V (x) + λ)φ(x) (8)φ + is positive and bounded on [A, +∞[ and φ − is positive and bounded on ] − ∞, −A]. (9)Let w λ := φ + (0)φ ′ −(0) − φ − (0)φ ′ +(0) and assume that ω λ ≠ 0. Then, for any positive and measurablefunction f on R and any x ∈ R:∫ +∞ [e −λt E x e − ∫ ]t0 V (Bs)ds f(B t ) dt = 2 ( ∫ x∫ +∞)φ + (x) φ − (y)f(y)dy + φ − (x) φ + (y)f(y)dy .0ω λ −∞x(10)Proof.We sketch the proof of this result for the sake of completeness. First, define the Wronskien:W λ (x) = φ + (x)φ ′ −(x) − φ − (x)φ ′ +(x).Since φ + and φ − are solutions of (8), we deduce that Wλ ′ (x) = 0, hence for any x ∈ R, W λ(x) = W λ (0) =ω λ . Assume first that f is continuous and has compact support, and define:∫ x∫ +∞φ(x) = φ + (x) φ − (y)f(y)dy + φ − (x) φ + (y)f(y)dy.−∞x9

φ is a function of C 1 -class, and differentiation yields:φ ′ (x) = φ ′ +(x)∫ x−∞We thus deduce that φ is of C 2 -class, and from (8):φ − (y)f(y)dy + φ ′ −(x)∫ +∞xφ + (y)f(y)dy.φ ′′ (x) = 2(V (x) + λ)φ(x) − W λ (x)f(x) = 2(V (x) + λ)φ(x) − ω λ f(x).Observe also that, since f is a function with compact support, the function φ is bounded on R. Considernow the processM t = e −λt−∫ t0 V (Bs)ds φ(B t ) + ω λ2∫ t0e −λu−∫ u0 V (Bs)ds f(B u )du.From Itô’s formula, this process is a local martingale and we have the estimate:|M t | ≤ sup |φ(x)| + ω λx∈R 2 sup |f(x)|x∈R∫ t0e −λu du ≤ sup |φ(x)| + ω λx∈R 2λ sup |f(x)|.x∈RTherefore M is uniformly bounded, i.e. M is a bounded martingale andφ(x) = E x [M 0 ] = E x [M ∞ ] = ω [∫ +∞]λ2 E x e −λu−∫ u0 V (Bs)ds f(B u )du .By a monotone class argument, the assumption on the continuity of f may be dropped, so Relation (10)is in fact valid for any positive and measurable function with compact support. Let now f by a positiveand measurable function, and consider the sequence of functionsf n (y) = f(y)1 {|y|≤n} .Since the f n have compact support, we may apply Relation (10) and write, for n large enough:∫ +∞ [e −λt E x e − ∫ ]t0 V (Bs)ds f(B t )1 {|Bt|≤n} dt = 2 ( ∫ x∫ n)φ + (x) φ − (y)f(y)dy + φ − (x) φ + (y)f(y)dy .0ω λ −nxWe finally end the proof by letting n → +∞ and applying the monotone convergence theorem thanks tothe Condition (9).03.2 Proof of Theorem 5We restrict our attention to the case x = 0, for which the formulae take a simpler form. To prove Theorem5, we first remove the drift thanks to the Cameron-Martin formula, which states that for every functionalF :[]E 0 [F (B s + νs, s ≤ t)] = E 0 e νBt− ν2 t2 F (Bs , s ≤ t) = e − ν2 t [2 E0 eνB tF (B s , s ≤ t) ] .Therefore the double Laplace transform reads:∫ +∞( ∫ t (e −(λ+a)t E 0[exp −α e 2β(Bs+νs) − 10∫ +∞=0ν2 −(λ+e 2 )t E 0[e νBt exp0(−α∫ tConsider the ordinary differential equation:12 φ′′ (x) =0) +ds − a∫ t(e 2β(Bs+νs) − 10() )] −e 2b(Bs+νs) − 1 ds dt) +ds − at − a∫ t(α(e 2βx − 1)1 {x>0} + a(e 2bx − 1)1 {x

The function V defined byV (x) = α(e 2βx − 1)1 {x>0} + a(e 2bx − 1)1 {x

where the Wronskien ω λ equals:ω λ = √ 2λ + ν 2 K 2γ(√2αβ)− √ (√ )2α2αK 2γ′ .βWe may also recover the Laplace transform of the associated perpetuity as follows. Assume that ν < 0and replace λ by λε to obtain:∫ +∞0e −λεt E 0[e −α ∫ t0 (e 2β(Bs+νs) −1) + ds ] dt = 1 εNow letting ε → 0, we deduce that:1λ E 0[e −α ∫ +∞0 (e 2β(Bs+νs) −1) ] ⎛+ ds 2ε= lim ⎝ε→0 w λε= 2|ν|λ∫ +∞0∫ +∞0e −λt E 0[e −α ∫ t/ε0 (e 2β(Bs+νs) −1) + ds ] dt.e νy K 2γ(√2αβ eβy )dy +( √2α )K 2γ β( √2α )|ν|K 2γ − √ ( √2α ),2αK 2γ ′ββ( √2α ) ⎞K 2γ βν + √ ⎠2λε + ν 2which agrees with [PRY10, Section 4.4, p.107]. Unfortunately, it does not seem easy to invert this Laplacetransform.4.2 The positive sojourn time of Brownian motion with driftLet first α → 0. From the asymptotics (see Section 5.1):we deduce that∫ +∞02 ν−1 Γ(ν)K ν (z) ∼z→0 z ν and zK ν(z) ′ ν2 ν−1 Γ(ν)= −zK ν−1 (z) − νI ν (z) ∼z→0 z ν[e −λt E 0 e −at−a ∫ t0 (e 2b(Bs+νs) −1) ] ⎛− dsdt = 2 ∫ 0(√ )⎝2ae νy I 2γω λ −∞ b eby dy +where the Wronskien ω λ equals:ω λ = √ 2aI ′ 2c(√ )2aWe now let further b → +∞. The left-hand side yields :t +∫ t0() −∫ te 2b(Bs+νs) − 1 ds = t +b−−−−→b→+∞+ √ (√ )2a2(λ + a) + ν 2 I 2c .be 2b(Bs+νs) 1 {Bs+νs

andI ′ √2λ+ν 2b(√ )2ab= I √ 2λ+ν 2b +1(√ )2a+b√2λ + ν2√2aI √ 2λ+ν 2b(√ )2ab−−−−→b→+∞√2λ + ν2√2a.Therefore, we deduce that∫ +∞0e −λt E 0[e −a ∫ t0 1 {Bs+νs>0}ds ] dt = 2ω λ( ∫ 0−∞e νy e y√ 2λ+ν 2 dy +)1√2(λ + a) + ν2 − ν)(= 2 1ω λ ν + √ 2λ + ν + 1√ 2 2(λ + a) + ν2 − ν,withWe now study two further simplifications :i) When ν = 0, this expression simplifies to:∫ +∞0ω λ = √ 2λ + ν 2 + √ 2(λ + a) + ν 2 .0[e −λt E 0 e −a ∫ ] t0 1 1{Bs>0}dsdt = √ √λ λ + aand this double Laplace transform may be inverted to recover the celebrated Arcsine law of Brownianmotion:(∫ t)1P 0 1 {Bs>0}ds ∈ dz =π √ z √ t − z 1 {00}ds ∈ dz = |ν|√ 2 1√ √ e − ν22 z − |ν| √ 2π z4.3 Yor’s functional0Take a = α and b = β. Then, from (11), the Wronskien simplifies toand from the formula (see [GR07, p.712]) :we obtain the expression :∫ +∞02I γ (x)K γ (y) =∫ +∞0ω λ = βe − t 2 − x2 +y 22te −λt E 0[e −α ∫ t0 e2β(Bs+νs) ds ] dtI γ( xyt∫ +∞|ν| √ z√2e −t2 dt) dt, for y ≥ x,t)1 {z>0} dz.13

= 1 β= 1 β∫ +∞0∫ +∞0dtt exp (− t 2 − αdzz e−αz exp(exp − α)tβ∫R2(− 1 )2zβ∫R2 exp) ( ) 2αetβ 2 e2βy e νy βyI 2γtβ 2 dy(− 1 ) ( ) e2zβ 2 e2βy e νy βyI 2γzβ 2 dy.We may therefore invert the Laplace transform in α to obtain:(∫ τλ)P 0 e 2β(Bs+νs) ds ∈ dz = λ ∫ (10zβ e− 2zβ 2 exp − 1 )( ) eR 2zβ 2 e2βy e νy βyI √ 2λ+ν 2β zβ 2 dywhere τ λ denotes an exponential random variable with parameter λ independent from B.Note that this last Laplace transform may be also inverted thanks to the Hartman-Watson function θ rgiven by:∫ +∞I √ 2λ (r) = e −λt θ r (t)dtwhich, from Yor [Yor80], admits the representation:θ r (t) =r √2π3 t∫ +∞4.4 One-sided Yor’s functionalTake a = α and let b → +∞. We obtain :∫ +∞0[e −λt E 0 e −α ∫ ] ⎛t0 e2β(Bs+νs) 1 {Bs+νs>0} dsdt = 2 ⎝ K 2γω λ00(e π2 −y 22t −r πy)cosh(y) sinh(y) sin dy.t( √2α)∫ +∞βν + √ 2λ + ν + 20(√ ) ⎞2αe νy K 2γβeβy ⎠with(√ )2α √2λω λ = K 2γ + ν2 − √ 2αK 2γ′ β(√ )2α.βThis allows to recover the associated perpetuity, for ν < 0 :(E 0[exp −α∫ +∞0)]e 2β(Bs+νs) 1 {Bs+νs>0}ds =2|ν|K |ν|β( √2αβ)√ ( √2α2αK |ν|β +1 βWe refer to Salminen & Yor [SY04, SY05] for a comprehensive study of this family of perpetuities.5 Appendix on Bessel functions and Bessel processes (with drift)5.1 Modified Bessel functions [Leb72, Chapter 5]For ν ∈ C, let I ν denote the modified Bessel function defined by:).I ν (x) =∞∑k=0(x/2) ν+2kΓ(k + 1)Γ(k + ν + 1)x > 0,and K ν the McDonald function defined, for ν /∈ Z, by:K ν (x) = π 2I −ν (x) − I ν (x)sin(νπ)x > 0,14

and for ν = n ∈ Z by :K n (x) = lim K ν (x).It is known that these functions generate the set of solutions of the linear differential equation:u ′′ + 1 )x u′ −(1 + ν2x 2 u = 0.ν→nν≠nTheir derivatives are seen to satisfy several recurrence relations:⎧⎪⎨ xI ν(x) ′ = xI ν−1 (x) − νI ν (x) = xI ν+1 (x) + νI ν (x),⎪⎩xK ν(x) ′ = −xK ν−1 (x) − νK ν (x) = −xK ν+1 (x) + νK ν (x),and their Wronskien takes a particularly simple form:W (I ν (x), K ν (x)) := I ν (x)K ′ ν(x) − I ′ ν(x)K ν (x) = − 1 x . (11)Note that they both simplify when ν = ±1/2:√ √22I −1/2 (z) =πz cosh(z), I 1/2(z) =πz sinh(z)andK −1/2 (z) = K 1/2 (z) =√ π2z e−z .For real and strictly positive ν > 0, I ν is a positive increasing function and K ν is a positive decreasingfunction. In this case, they both admit some useful integral representation:⎧x ν ∫ πI ν (x) =⎪⎨ 2 ν√ π Γ(ν + 1 2 ) e x cos(θ) sin 2ν (θ)dθ0⎪⎩ K ν (x) = 1 ( x) ν∫ +∞( )12 2 t ν+1 exp −t − x2dt4tfrom which we may deduce the following equivalents (still for ν > 0):0x νI ν (x) ∼x→0 2 ν Γ(ν + 1)and the asymptotic formulae when x → +∞:andK ν (x) ∼x→02 ν−1 Γ(ν)x ν ,I ν (x)∼x→+∞5.2 Bessel processes with drift [PY81]√e xπ√ and K ν (x) ∼2πx x→+∞ 2x e−x .Let δ ∈ N\{0} and c > 0. The Bessel process of dimension δ (or equivalently of index ν = δ 2− 1) and driftc is the diffusion (R (δ,c)t , t ≥ 0) with generator:G = 1 ∂ 2 ( 2ν + 12 ∂x 2 + + c I )ν+1 ∂(cx)2x I ν ∂x . (12)15

We denote by P (δ,c)x its law when started from x. This process may be obtained as the euclidean norm ofa δ-dimensional Brownian motion −→ B with drift −→ µ ∈ R δ such that ‖ −→ µ ‖ = c:R (δ,c)t= ‖ −→ B t + −→ µ −→ t ‖.The law of the first passage times of (R t , t ≥ 0) is given by the following Laplace transform:E (δ,c)xI ν (x⎧⎪ √ 2λ + c 2 )[e−λT a] ⎨ I ν (a √ I ν (ca)2λ + c 2 ) I ν (cx)=⎪ ⎩K ν (x √ 2λ + c 2 )K ν (a √ I ν (ca)2λ + c 2 ) I ν (cx)In particular, when δ = 3, (i.e. ν = 1 2), these formulae simplify to:E (3,c)xsinh(x⎧⎪ √ 2λ + c 2 )[e−λT a] ⎨ sinh(a √ sinh(ca)2λ + c 2 ) sinh(cx)=⎪ ⎩(exp −(x − a) √ ) sinh(ca)2λ + c 2 sinh(cx)Note that this Laplace transform may be inverted, see [EMOT54, p.258]:P (3,c)x (T a ∈ dt)/dt =⎧sinh(ca) 1√sinh(cx) 2πt3⎪⎨sinh(ca)⎪⎩sinh(cx)5.3 Bessel processes+∞∑n=−∞((2n + 1)a − x) exp()(x − a)√ exp (x − a)2− − c22πt3 2t 2 t(−if x ≤ a,if x ≥ a.if x ≤ a,if x ≥ a.((2n + 1)a − x)22t− c2 2 t )Letting c → 0 informally in (12), we obtain the generator of the classic Bessel process with index ν:if x ≤ a,if x ≥ a.G = 1 2 ∂x 2 + 2ν + 1 ∂2x ∂x . (13)These processes are deeply related with geometric Brownian motion thanks to the following Lamperti’srelation:∂ 2Theorem 7 ([Wil74]). Let (B t + νt, t ≥ 0) be a Brownian motion with drift ν > 0 starting from log(x).Then, there exists (R (ν)t , t ≥ 0) a Bessel process of index ν started from x such that:exp(B t + νt) = R (ν) ∫ t0 exp(2(Bs+νs))ds.In particular, for ν > 0, (R (ν)t , t ≥ 0) is transient and limt→+∞ R(ν) t = +∞ a.s. In the framework ofperpetuities, we may mention the following theorem, which allows to compute the law of some perpetuitiesinvolving squared Bessel processes, see [RY99, Theorem 1.7 p.444]:Theorem 8. Let ϕ be a positive and measurable function such that ∫ +∞(1 + x)ϕ(x)dx < +∞. Then:0[ ( ∫ +∞)]( x)exp − Rt 2 ϕ(t)dt = F (+∞) δ/2 exp2 F ′ (0)E (δ)xwhere F is the unique solution on [0, +∞[ of :0F ′′ = ϕ(x)F such that F is positive, non increasing, and F (0) = 1.16

5.4 The three-dimensional Bessel process [RY99, Chapter VI.3]We now take δ = 3 (i.e. ν = 1 2) in (13) to obtain the classic three-dimensional Bessel process. This processenjoys many important properties and is, in some sense, very close de Brownian motion. In particular,there is a weak absolute continuity formula between Brownian motion and the three-dimensional Besselprocess :P (3)x|F t= B tx 1 {t>T 0} P x|Ft . (14)The paths of a three-dimensional Bessel process admit a useful decomposition as follows:Theorem 9. Let (R t , t ≥ 0) be a three-dimensional Bessel process started from 0 and define G x = sup{t ≥0, R t = x} it last passage time at level x. Then:i) conditionally on G x , the processes (R t , t ≤ G x ) and (R t+Gx , t ≥ 0) are independent,ii) the process (R t+Gx , t ≥ 0) has the same law as (x + R t , t ≥ 0),iii) the process (R t , t ≤ G x ) has the same law as (B T0−t, t ≤ T 0 ) where (B t , t ≥ 0) is a Brownian motionstarted from x and T 0 = inf{t ≥ 0, B t = 0}.This result was first proven by Williams [Wil74] in its decomposition of the Brownian paths, see alsoPitman [Pit75] for a related study.We conclude this appendix by stating the third Ray-Knight theorem, which describes the dependence inthe space variable of the total local time of the three-dimensional Bessel process :Theorem 10 (Ray-Knight). Let (R t , t ≥ 0) be a three-dimensional Bessel process started from 0 anddenote by L y ∞(R) its total local time at level y. Then:(L y ∞, y ≥ 0) (law)= (Z (2)y , y ≥ 0)where Z (2) is a two-dimensional squared Bessel process started from 0.We refer to [BS02, Chapter V] for other similar Ray-Knight theorems.References[AS92][BS02]M. Abramowitz and I. A. Stegun, editors. Handbook of mathematical functions with formulas,graphs, and mathematical tables. Dover Publications Inc., New York, 1992. Reprint of the1972 edition.A. N. Borodin and P. Salminen. Handbook of Brownian motion—facts and formulae. Probabilityand its Applications. Birkhäuser Verlag, Basel, second edition, 2002.[Cha12] D. Chamorro. Algunas herramientas matemáticas para la economía y las finanzas: elmovimiento Browniano y la integral de Wiener. Analítika, 3:3–15, 2012.[DDSGS05] M. Decamps, A. De Schepper, M. Goovaerts, and W. Schoutens.perpetuities. Scand. Actuar. J., (4):261–270, 2005.A note on some new[Duf90][EMOT54]D. Dufresne. The distribution of a perpetuity, with applications to risk theory and pensionfunding. Scand. Actuar. J., (1-2):39–79, 1990.A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Tables of integral transforms.Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part,on notes left by Harry Bateman.17

[GR07]I. S. Gradshteyn and I. M. Ryzhik. Table of integrals, series, and products. Elsevier/AcademicPress, Amsterdam, seventh edition, 2007. Translated from the Russian, Translation editedand with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows,Macintosh and UNIX).[GY93] H. Geman and M. Yor. Bessel processes, Asian options and perpetuities. Mathematicalfinance, 3(4):349–375, 1993.[Jan07][Leb72][Pit75]S. Janson. Brownian excursion area, Wright’s constants in graph enumeration, and otherBrownian areas. Probab. Surv., 4:80–145, 2007.N. N. Lebedev. Special functions and their applications. Dover Publications Inc., New York,1972. Revised edition, translated from the Russian and edited by Richard A. Silverman,Unabridged and corrected republication.J. W. Pitman. One-dimensional Brownian motion and the three-dimensional Bessel process.Advances in Appl. Probability, 7(3):511–526, 1975.[PRY10] C. Profeta, B. Roynette, and M. Yor. Option prices as probabilities. Springer Finance.Springer-Verlag, Berlin, 2010. A new look at generalized Black-Scholes formulae.[PY81]J. Pitman and M. Yor. Bessel processes and infinitely divisible laws. In Stochastic integrals(Proc. Sympos., Univ. Durham, Durham, 1980), volume 851 of Lecture Notes in Math., pages285–370. Springer, Berlin, 1981.[PZ03] A. D. Polyanin and V. F. Zaitsev. Handbook of exact solutions for ordinary differentialequations. Chapman & Hall/CRC, Boca Raton, FL, second edition, 2003.[RY99] D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of MathematicalSciences]. Springer-Verlag, Berlin, third edition, 1999.[SY04]P. Salminen and M. Yor. On Dufresne’s perpetuity, translated and reflected. In Stochasticprocesses and applications to mathematical finance, pages 337–354. World Sci. Publ., RiverEdge, NJ, 2004.[SY05] P. Salminen and M. Yor. Perpetual integral functionals as hitting and occupation times.Electron. J. Probab., 10:no. 11, 371–419 (electronic), 2005.[Wil74][Yor80]D. Williams. Path decomposition and continuity of local time for one-dimensional diffusions.I. Proc. London Math. Soc. (3), 28:738–768, 1974.M. Yor. Loi de l’indice du lacet brownien, et distribution de Hartman-Watson. Z. Wahrsch.Verw. Gebiete, 53(1):71–95, 1980.18