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106 CHAPTER 3. NUMERICAL IMPLEMENTATION discrepancy principle cannot be applied are given in [95] and [50]. However, in our numerical tests (see Section 3.6) we have always been able to find a solution to (3.13) We will now give a simple sufficient condition, adapted from [94], under which (3.13) admits a solution. Proposition 3.4. Suppose that the hypotheses 1–3 of page 103 are satisfied and let c 1 and c 2 satisfy (3.12). If ε δ (α) is a single-valued function then there exists an α satisfying (3.13). This proposition is a direct consequence of the following lemma. Lemma 3.5. The function ε δ (α) is non-decreasing and satisfies the following limit relations: lim ε δ (α) ≤ δ 2 , α↓0 lim ε δ(α) = ‖C Q∗ − C δ α→∞ M‖ 2 w. If, at some point α > 0, ε δ (α) is single-valued, then it is continuous at this point. The function J δ (α) := ‖C Qδ α − CM‖ δ 2 w + αI(Q δ α|P ). is non-decreasing, continuous, and satisfies the following limit relations: lim J δ (α) ≤ δ 2 , α↓0 lim J δ(α) ≥ ‖C Q∗ − C δ α→∞ M‖ 2 w. Proof. Let γ δ (α) := I(Q δ α|P ) and let 0 < α 1 < α 2 . By the optimality of Q δ α 1 and Q δ α 2 we have: and therefore ε δ (α 1 ) + α 1 γ δ (α 1 ) ≤ ε δ (α 2 ) + α 1 γ δ (α 2 ), ε δ (α 2 ) + α 2 γ δ (α 2 ) ≤ ε δ (α 1 ) + α 2 γ δ (α 1 ) ε δ (α 2 ) − ε δ (α 1 ) ≥ α 1 (γ δ (α 1 ) − γ δ (α 2 )) ε δ (α 2 ) − ε δ (α 1 ) ≤ α 2 (γ δ (α 1 ) − γ δ (α 2 )), which implies that ε δ (α 2 ) ≥ ε δ (α 1 ) and γ δ (α 1 ) ≥ γ δ (α 2 ). To prove the first limit relation for ε δ (α), observe that for all α > 0, ε δ (α) ≤ ‖C Q+ − C δ M‖ 2 w + αI(Q + |P ) ≤ δ 2 + αI(Q + |P ) −−−→ α→0 δ2 . To prove the second limit relation for ε δ (α), one can write, using the optimality of Q δ α: ε δ (α) + αγ δ (α) ≤ ‖C Q∗ − C δ M‖ 2 w + αI(Q ∗ |P ).
3.3. CHOICE OF THE REGULARIZATION PARAMETER 107 From [30, Theorem 2.2], γ δ (α) ≥ I(Q δ α|Q ∗ ) + I(Q ∗ |P ). (3.16) Therefore Using the inequality I(Q δ α|Q ∗ ) ≤ ‖CQ∗ − C δ M ‖2 w α −−−→ α→∞ 0. |P − Q| ≤ √ 2I(P |Q), (3.17) where |P − Q| denotes the total variation distance (see [30, Equation (2.3)]), this implies that Q δ α converges to Q ∗ in total variation distance (and therefore also weakly) as α goes to infinity. The limit relation now follows from Lemma 2.2. To prove the continuity of ε δ (α), let {α n } be a sequence of positive numbers, converging to α > 0. By the optimality of Q δ α n , I(Q δ α n |P ) is bounded and one can choose a subsequence of {Q δ α n }, converging weakly toward some measure Q ′ , and denoted, to simplify notation, again by {Q δ α n } n≥1 . We now need to prove that Q ′ is the solution of the calibration problem with regularization parameter α. By weak continuity of the pricing error (Lemma 2.2) and weak lower semicontinuity of the relative entropy (Lemma 2.11), we have for any other measure Q: ‖C Q′ − C δ M‖ 2 w + αI(Q ′ |P ) ≤ lim inf n {‖CQδ αn − C δ M ‖ 2 w + αI(Q δ α n |P )} = lim inf n {‖CQδ αn − C δ M ‖ 2 w + α n I(Q δ α n |P ) + (α − α n )I(Q δ α n |P )} ≤ lim inf n {‖CQ − C δ M‖ 2 w + α n I(Q|P ) + (α − α n )I(Q δ α n |P )} = ‖C Q − C δ M‖ 2 w + αI(Q|P ). Therefore, the sequence {ε δ (α n )} converges to one of the possible values of ε δ (α). If this function is single-valued in α, this means that every subsequence of the original sequence {ε δ (α n )} has a further subsequence that converges toward ε δ (α), and therefore, the original sequence also converges toward ε δ (α). The fact that J δ (α) is nondecreasing as a function of α is trivial. To show the continuity, observe that for 0 < α 1 < α 2 , J δ (α 2 ) − J δ (α 1 ) ≤ ε δ (α 1 ) + α 2 γ δ (α 1 ) − ε δ (α 1 ) − α 1 γ δ (α 1 ) = (α 2 − α 1 )γ δ (α 1 ) ≤ (α 2 − α 1 )I(Q + |P ).
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Thèse présentée pour obtenir le
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Abstract This thesis deals with the
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Contents Remarks on notation 11 Int
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CONTENTS 9 II Multidimensional mode
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12 For {P n } n≥1 , P ∈ P(Ω) t
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14 INTRODUCTION EN FRANCAIS corresp
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16 INTRODUCTION EN FRANCAIS Lévy p
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18 INTRODUCTION EN FRANCAIS Calibra
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20 INTRODUCTION EN FRANCAIS • Si
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22 INTRODUCTION EN FRANCAIS et Scai
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24 INTRODUCTION EN FRANCAIS Ce rés
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26 INTRODUCTION EN FRANCAIS dans un
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28 INTRODUCTION 7500 Taux de change
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30 INTRODUCTION Lévy models by Fou
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Chapter 1 Lévy processes and exp-L
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1.1. LEVY PROCESSES 35 Proposition
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1.1. LEVY PROCESSES 37 The characte
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1.1. LEVY PROCESSES 39 On the other
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1.2. EXPONENTIAL LEVY MODELS 41 (b)
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1.3. EXP-LEVY MODELS IN FINANCE 43
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1.3. EXP-LEVY MODELS IN FINANCE 45
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1.4. PRICING EUROPEAN OPTIONS 47 wi
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1.4. PRICING EUROPEAN OPTIONS 49 Pr
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1.4. PRICING EUROPEAN OPTIONS 51 Nu
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1.4. PRICING EUROPEAN OPTIONS 53 By
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Page 115 and 116: 3.5. NUMERICAL SOLUTION 115 of X t
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Page 131: Part II Multidimensional modelling
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156 CHAPTER 4. DEPENDENCE OF LEVY P
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166 CHAPTER 5. APPLICATIONS OF LEVY
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188 CONCLUSIONS AND PERSPECTIVES ge
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190 BIBLIOGRAPHY [9] O. E. Barndorf
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192 BIBLIOGRAPHY [33] F. Delbaen, P
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194 BIBLIOGRAPHY [57] P. Jorion, On
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196 BIBLIOGRAPHY [81] S. Raible, L
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198 BIBLIOGRAPHY
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200 INDEX tightness, 40 levycalibra
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