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142 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES Definition 4.5 (F-volume). Let F : S ⊂ ¯R d → ¯R. For a, b ∈ S, with a ≤ b, the F -volume of |a, b| is defined by V F (|a, b|) := ∑ (−1) N(c) F (c), (4.4) where the sum is taken over all vertices c of |a, b|, and N(c) := #{k : c k = a k }. The notion of F-volume should only be seen as a convenient notation for the sum in the right-hand side of (4.4). It does not in general correspond to any measure because the measure of |a, b| will depend on whether the boundary is included or not. In particular, in two dimensions (d = 2) the F -volume of a rectangle B = |x 1 , x 2 | × |y 1 , y 2 | satisfies V F (B) = F (x 2 , y 2 ) − F (x 2 , y 1 ) − F (x 1 , y 2 ) + F (x 1 , y 1 ). If F (u) = ∏ d i=1 u i, the F -volume of any interval is equal to its Lebesgue measure. Definition 4.6 (d-increasing function). A function F : S ⊂ ¯R d → ¯R is called d-increasing if for all a, b ∈ S with a ≤ b, V F (|a, b|) ≥ 0. Definition 4.7 (Grounded function). For each k, let S k ⊂ ¯R be such that inf S k ∈ S k . A function F : S 1 × · · · × S d → ¯R is grounded if F (x 1 , . . . , x d ) = 0 whenever x k = inf S k for at least one k. Definition 4.8 (margins of a d-increasing grounded function). For each k, let S k ⊂ ¯R be such that inf S k ∈ S k and sup S k ∈ S k and let F : S 1 × · · · × S d → ¯R be d-increasing and grounded. Then for I ⊂ {1, . . . , d} nonempty, the I-margin of F is a function F I : ∏ i∈I S i → ¯R defined by F I ((x i ) i∈I ) := F (x 1 , . . . , x d )| xi =sup S i , i∈Ic. (4.5) When I = {k}, to simplify notation, the (one-dimensional) I-margin of F is denoted by F k . Example 4.3. The distribution function F of a random vector X ∈ R d is usually defined by F (x 1 , . . . , x d ) := P [X 1 ≤ x 1 , . . . , X d ≤ x d ]
4.3. INCREASING FUNCTIONS 143 for x 1 , . . . , x d ∈ ¯R. F is then clearly increasing because for every a, b ∈ ¯R d with a ≤ b, V F (|a, b|) = P [X ∈ (a, b]] (4.6) It is grounded because F (x 1 , . . . , x d ) = 0 if x k = −∞ for some k, and the margins of F are the distribution functions of the margins of X: for example, F 1 (x) = F (x, ∞, . . . , ∞) = P [X 1 ≤ x]. The following technical lemma will be useful in the sequel. Lemma 4.3. For each k, let S k ⊂ ¯R be such that inf S k ∈ S k and let F, H : S 1 × · · · × S d → ¯R be d-increasing and grounded. Then their product F H is d-increasing and grounded. Proof. We will prove this lemma by induction on d. The product of two increasing grounded functions on ¯R is clearly increasing and grounded. Suppose d ≥ 2, and for each k let a k , b k ∈ S k with a k ≤ b k . Consider the function ˜F (u 2 , . . . , u d ) := F (b 1 , u 2 , . . . , u d )H(b 1 , u 2 , . . . , u d ) − F (a 1 , u 2 , . . . , u d )H(a 1 , u 2 , . . . , u d ) = H(a 1 , u 2 , . . . , u d )[F (b 1 , u 2 , . . . , u d ) − F (a 1 , u 2 , . . . , u d )] + F (b 1 , u 2 , . . . , u d )[H(b 1 , u 2 , . . . , u d ) − H(a 1 , u 2 , . . . , u d )] Since H is grounded, V H(a1 ,∗)(|a 2 , b 2 | × · · · × |a d , b d |) = V H (|0, a 1 | × |a 2 , b 2 | × · · · × |a d , b d |) ≥ 0, V H(b1 ,∗)−H(a 1 ,∗)(|a 2 , b 2 | × · · · × |a d , b d |) = V H (|a 1 , b 1 | × |a 2 , b 2 | × · · · × |a d , b d |) ≥ 0, and the same is true for F . Therefore ˜F is increasing by the induction hypothesis. Since V F H (|a 1 , b 1 | × · · · × |a d , b d |) = V ˜F (|a 2 , b 2 | × · · · × |a d , b d |), this finishes the proof of the lemma. To develop the theory of Lévy copulas for general Lévy processes, we will need to define the notion of margins for a function that is not grounded. The following example gives an intuition of how this can be done. Example 4.4. Consider the following “alternative” definition of a distribution function of a random vector X: d∏ ˜F (x 1 , . . . , x d ) := P [X 1 ∈ (x 1 ∧ 0, x 1 ∨ 0]; . . . ; X d ∈ (x d ∧ 0, x d ∨ 0]] sgn x i (4.7) i=1
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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 99 and 100: 3.2. CHOICE OF THE PRIOR 99 Since
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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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Page 188 and 189: 188 CONCLUSIONS AND PERSPECTIVES ge
Page 190 and 191: 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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