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146 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES Fix ε > 0 and choose a i , b i ∈ S i : a i ≤ b i , i = 2, . . . , d such that F 1 (y 1 ) ≤ ε + V F (|x ∗ 1, y 1 | × |a 2 , b 2 | × · · · × |a d , b d |). For i = 2, . . . , d let ã i := a i ∧ x ∗ i and ˜b i := b i ∨ x i and denote ˜B := |ã 2 , ˜b 2 | × · · · × |ã d , ˜b d |. Then, since F is d-increasing, V F (|y 1 , x 1 | × B) ≤ V F (|y 1 , x 1 | × ˜B) and V F (|x ∗ 1 , y 1| × |a 2 , b 2 | × · · · × |a d , b d |) ≤ V F (|x ∗ 1 , y 1| × ˜B). Therefore V F (|y 1 , x 1 | × B) ≤ V F (|x ∗ 1, x 1 | × ˜B) − F 1 (y 1 ) + ε ≤ F 1 (x 1 ) − F 1 (y 1 ) + ε. Since the above is true for all ε > 0, in view of (4.12), the proof of (4.11) is complete. We close this section with the definition of (ordinary) copula and the Sklar’s theorem, which relates copulas to distribution functions. The proof of Sklar’s theorem can be found in [90]. Definition 4.11 (Copula). A d-dimensional copula (a d-copula) is a function C : [0, 1] d → [0, 1] such that 1. C is grounded and d-increasing. 2. C has margins C k , k = 1, 2, . . . , d, which satisfy C k (u) = u for all u in [0, 1]. Theorem 4.5 (Sklar). Let F be a d-dimensional distribution function with margins F 1 , . . . , F d . Then there exists a d-dimensional copula C such that for all x ∈ ¯R d , F (x 1 , x 2 , . . . , x d ) = C(F 1 (x 1 ), F 2 (x 2 ), . . . , F d (x d )). (4.13) If F 1 , . . . , F d are continuous then C is unique; otherwise, C is uniquely determined on Ran F 1 × · · · × Ran F d . Conversely, if C is a d-copula and F 1 , . . . , F d are distribution functions, then the function F defined by (4.13) is a d-dimensional distribution function with margins F 1 , . . . , F d . 4.4 Lévy copulas for spectrally positive Lévy processes This section discusses the notion of Lévy copula for Lévy processes with only positive jumps in each component. This notion was introduced by the present author in [92]. Examples of Lévy copulas for spectrally positive Lévy processes and methods to construct them will be given in Sections 4.6 and 5.1 together with the examples of general Lévy copulas. Further properties of
4.4. LEVY COPULAS: SPECTRALLY POSITIVE CASE 147 Lévy copulas in the spectrally positive case, including a characterization of the convergence of Lévy measures in terms of Lévy copulas can be found in a recent paper by Barndorff-Nielsen and Lindner [8]. As the laws of random variables are represented by their distribution functions, Lévy measures can be represented by their tail integrals. Definition 4.12. Let ν be a Lévy measure on R d + := [0, ∞) d \ {0}. The tail integral U of ν is a function [0, ∞) d → [0, ∞] such that 1. U(0, . . . , 0) = ∞. 2. For (x 1 , . . . , x d ) ∈ R d +, U(x 1 , . . . , x d ) = ν([x 1 , ∞) × · · · × [x d , ∞)). The Lévy measure is uniquely determined by its tail integral, because the above definition implies that for every x, y ∈ R d + with x ≤ y, V U (|x, y|) = (−1) d ν([x 1 , y 1 ) × · · · × [x d , y d )). (4.14) Definition 4.13. Let X be a R d -valued Lévy process and let I ⊂ {1, . . . , d} non-empty. The I-marginal tail integral U I of X is the tail integral of the process X I := (X i ) i∈I . The onedimensional margins are, as usual, denoted by U i := U {i} . Lemma 4.1 entails that for I ⊂ {1, . . . , d} nonempty, the I-marginal tail integral of a Lévy measure ν on R d + can be computed from the tail integral U of ν by substituting 0 instead of arguments with indices not in I: U I ((x i ) i∈I ) = U(x 1 , . . . , x d )| (xi ) i∈I c =0. (4.15) A Lévy copula is defined similarly to ordinary copula but on a different domain. Definition 4.14. A function F : [0, ∞] d → [0, ∞] is a Lévy copula if 1. F (u 1 , . . . , u d ) < ∞ for (u 1 , . . . , u d ) ≠ (∞, . . . , ∞), 2. F is grounded: F (u 1 , . . . , u d ) = 0 whenever u i = 0 for at least one i ∈ {1, . . . , d},
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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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Page 115 and 116: 3.5. NUMERICAL SOLUTION 115 of X t
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Page 190 and 191: 190 BIBLIOGRAPHY [9] O. E. Barndorf
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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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