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176 CHAPTER 5. APPLICATIONS OF LEVY COPULAS Lemma 5.5. Let F be a Lévy copula on (−∞, ∞] d , satisfying (5.5), and F ξ be the corresponding conditional distribution function, defined by (5.8). Then, there exists N ⊂ R with λ(N) = 0 such that for every fixed ξ ∈ R \ N, F ξ (∗) is a probability distribution function, satisfying F ξ (x 2 , . . . , x d ) = sgn(ξ) ∂ ∂ξ V F ((ξ ∧ 0, ξ ∨ 0] × (−∞, x 2 ] × · · · × (−∞, x d ]) (5.9) in every point (x 2 , . . . , x d ), where F ξ is continuous. Remark 5.2. Since the law of a random variable is completely determined by the values of its distribution function at the continuity points of the latter, being able to compute F ξ at all points where it is continuous is sufficient for all practical purposes. Proof. Since it has already been observed that K(ξ, R d−1 ) = 1 λ-almost everywhere, we only need to prove the second part of the lemma. Let G(x 1 , . . . , x d ) := sgn x 1 V F ((x 1 ∧ 0, x 1 ∨ 0] × (−∞, x 2 ] × · · · × (−∞, x d ]) By Theorem 2.28 in [1], for each f ∈ L 1 (R d , µ), ∫ ∫ ∞ f(x 1 , . . . , x d )µ(dx 1 · · · dx d ) = dx 1 f(x 1 , . . . , x d )K(x 1 , dx 2 · · · dx d ), (5.10) R d −∞ ∫R d which implies that ∫ G(x 1 , . . . , x d ) = sgn x 1 (x 1 ∧0,x 1 ∨0] dξF ξ (x 2 , . . . , x d ), Therefore, for fixed (x 2 , . . . , x d ), (5.9) holds ξ-almost everywhere. Since a union of countably many sets of zero measure is again a set of zero measure, there exists a set N ⊂ R with λ(N) = 0 such that for every ξ ∈ R \ N, (5.9) holds for all (x 2 , . . . , x d ) ∈ Q d , where Q denotes the set of rational numbers. Fix ξ ∈ R \ N and let x ∈ R d−1 and {x + n } and {x − n } be two sequences of d − 1-dimensional vectors with coordinates in Q, converging to x from above and from below (componentwise). Since F ξ is increasing in each coordinate (as a probability distribution function), the limits lim n F ξ (x + n ) and lim n F ξ (x − n ) exist. Suppose that lim F ξ (x + n n ) = lim F ξ (x − n n ) = F ∗ (5.11) and observe that for every δ ≠ 0, G(ξ + δ, x − n ) − G(ξ, x − n ) δ ≤ G(ξ + δ, x) − G(ξ, x) δ ≤ G(ξ + δ, x+ n ) − G(ξ, x + n ) . δ
5.2. SIMULATION OF DEPENDENT LEVY PROCESSES 177 For every ε > 0, in view of (5.11), there exists N 0 such that for every n ≥ N 0 , F ξ (x + n )−F ∗ ≤ ε/2 and F ∗ − F ξ (x − n ) ≤ ε/2. Since G is differentiable with respect to the first variable at points (ξ, x + n ) and (ξ, x − n ), we can choose δ small enough so that and This proves that G(ξ + δ, x − n ) − G(ξ, x − n ) ∣ δ G(ξ + δ, x + n ) − G(ξ, x + n ) ∣ δ − F ξ (x − n ) ∣ ≤ ε/2 − F ξ (x + n ) ∣ ≤ ε/2 G(ξ + δ, x) − G(ξ, x) lim = F ∗ . δ→0 δ We have thus shown that F ξ satisfies Equation (5.9) in all points where (5.11) holds, that is, where F ξ is continuous. In the following two theorems we show how Lévy copulas may be used to simulate multidimensional Lévy processes with specified dependence. Our results can be seen as an extension to Lévy processes, represented by Lévy copulas, of the series representation results, developed by Rosinski and others (see [84] and references therein). The first result concerns the simpler case when the Lévy process has finite variation on compacts. Theorem 5.6. (Simulation of multidimensional Lévy processes, finite variation case) Let ν be a Lévy measure on R d , satisfying ∫ (|x| ∧ 1)ν(dx) < ∞, with marginal tail integrals U i , i = 1, . . . , d and Lévy copula F (x 1 , . . . , x d ), such that the condition (5.5) is satisfied, and let K(x 1 , dx 2 · · · dx d ) be the corresponding conditional probability distributions, defined by (5.8). Let {V i } be a sequence of independent random variables, uniformly distributed on [0, 1]. Introduce d random sequences {Γ 1 i }, . . . , {Γd i }, independent from {V i} such that • N = ∑ ∞ i=1 δ {Γ 1 i } is a Poisson random measure on R with intensity measure λ. • Conditionally on Γ 1 i , the random vector (Γ2 i , . . . , Γd i ) is independent from Γk j and all k and is distributed on R d−1 with law K(Γ 1 i , dx 2 · · · dx d ). with j ≠ i Then {Z t } 0≤t≤1 where Z k t = ∞∑ i=1 U (−1) i (Γ k i )1 [0,t] (V i ), k = 1, . . . , d, (5.12)
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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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1.4. PRICING EUROPEAN OPTIONS 55 Si
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Chapter 2 The calibration problem a
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2.1. LEAST SQUARES CALIBRATION 59 m
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2.1. LEAST SQUARES CALIBRATION 61 i
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2.1. LEAST SQUARES CALIBRATION 63 w
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2.1. LEAST SQUARES CALIBRATION 65 A
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2.1. LEAST SQUARES CALIBRATION 67 i
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2.2. SELECTION USING RELATIVE ENTRO
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2.2. SELECTION USING RELATIVE ENTRO
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2.3. RELATIVE ENTROPY IN THE LITERA
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2.4. RELATIVE ENTROPY OF LEVY PROCE
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2.4. RELATIVE ENTROPY OF LEVY PROCE
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2.5. REGULARIZING THE CALIBRATION P
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Chapter 3 Numerical implementation
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3.1. DISCRETIZING THE CALIBRATION P
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3.2. CHOICE OF THE PRIOR 99 Since
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3.2. CHOICE OF THE PRIOR 101 twice
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3.3. CHOICE OF THE REGULARIZATION P
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3.4. CHOICE OF THE WEIGHTS 111 Then
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3.5. NUMERICAL SOLUTION 113 have th
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3.5. NUMERICAL SOLUTION 115 of X t
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3.5. NUMERICAL SOLUTION 117 since
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3.6. NUMERICAL AND EMPIRICAL TESTS
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Page 125 and 126: 3.6. NUMERICAL AND EMPIRICAL TESTS
Page 131: Part II Multidimensional modelling
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Page 188 and 189: 188 CONCLUSIONS AND PERSPECTIVES ge
Page 190 and 191: 190 BIBLIOGRAPHY [9] O. E. Barndorf
Page 192 and 193: 192 BIBLIOGRAPHY [33] F. Delbaen, P
Page 194 and 195: 194 BIBLIOGRAPHY [57] P. Jorion, On
Page 196 and 197: 196 BIBLIOGRAPHY [81] S. Raible, L
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Page 200: 200 INDEX tightness, 40 levycalibra
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