Processus de LÃ©vy en Finance - Laboratoire de ProbabilitÃ©s et ...

More documents

Recommendations

Info

140 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES Now from the uniqueness of Lévy-Khintchine representation we conclude that ˜ν is the Lévy measure of X. The complete dependence of Lévy processes is a new notion that is worth being discussed in detail. First, the following definition is in order. Definition 4.2. A subset S of R d is called ordered if, for any two vectors v, u ∈ S, either v k ≤ u k , k = 1, . . . , d or v k ≥ u k , k = 1, . . . , d. S is called strictly ordered if, for any two different vectors v, u ∈ S, either v k u k , k = 1, . . . , d. We recall that random variables Y 1 , . . . , Y d are said to be completely dependent or comonotonic if there exists a strictly ordered set S ⊂ R d such that (Y 1 , . . . , Y d ) ∈ S with probability 1. However, saying that the components of a Lévy process are completely dependent only if they are completely dependent for every fixed time is too restrictive; the components of a Lévy process can be completely dependent as processes without being completely dependent as random variables for every fixed time. The following example clarifies this point. Example 4.2 (Dynamic complete dependence for Lévy processes). Let {X t } t≥0 be a Lévy process with characteristic triplet (A, ν, γ) such that A = 0 and γ = 0 and let {Y t } t≥0 be a Lévy process, constructed from the jumps of X: Y t = ∑ s≤t ∆X3 s . From the dynamic point of view X and Y are completely dependent, because the trajectory of any one of them can be reconstructed from the trajectory of the other. However, the copula of X t and Y t is not that of complete dependence because Y t is not a deterministic function of X t . Indeed, if X is a compound Poisson process having jumps of size 1 and 2 and X t = 3 for some t, this may either mean that X has three jumps of size 1 in the interval [0, t], and then Y t = 3, or that X has one jump of size 1 and one jump of size 2, and then Y t = 9. This example motivates the following definition. In this definition and below, K := {x ∈ R d : sgn x 1 = · · · = sgn x d }. (4.3) Definition 4.3. Let X be a R d -valued Lévy process. Its jumps are said to be completely dependent or comonotonic if there exists a strictly ordered subset S ⊂ K such that ∆X t := X t − X t− ∈ S, t ≥ 0 (except for a set of paths having zero probability).
4.3. INCREASING FUNCTIONS 141 Clearly, an element of a strictly ordered set is completely determined by one coordinate only. Therefore, if the jumps of a Lévy process are completely dependent, the jumps of all components can be determined from the jumps of any single component. If the Lévy process has no continuous martingale part, then the trajectories of all components can be determined from the trajectory of any one component, which indicates that Definition 4.3 is a reasonable dynamic notion of complete dependence for Lévy processes. The condition ∆X t ∈ K means that if the components of a Lévy process are comonotonic, they always jump in the same direction. For any R d -valued Lévy process X with Lévy measure ν and for any B ∈ B(R d \ {0}) the number of jumps in the time interval [0, t] with sizes in B is a Poisson random variable with parameter tν(B). Therefore, Definition 4.3 can be equivalently restated in terms of the Lévy measure ν of X as follows: Definition 4.4. Let X be a R d -valued Lévy process with Lévy measure ν. Its jumps are said to be completely dependent or comonotonic if there exists a strictly ordered subset S of K such that ν(R d \ S) = 0. 4.3 Increasing functions In this section we sum up some aspects of the theory of increasing functions of several variables. Most definitions are well known (see, e.g. [88]) but some are introduced either here for the first time or in [59]. Let ¯R := R ∪ {−∞} ∪ {∞} denote the extended real line. In the sequel, for a, b ∈ ¯R d such that a ≤ b (inequalities like this one are to be interpreted componentwise), let [a, b) denote a right-open left-closed interval of ¯R d (d-box): [a, b) := [a 1 , b 1 ) × · · · × [a d , b d ), In the same way we define other types of intervals: (a, b], [a, b] and (a, b). To simplify notation it is convenient to introduce a “general” type of interval: when we want to consider a d-dimensional interval as a set of its vertices, without specifying whether the boundary is included or not, we will write |a, b|.
Page 1 and 2:
Thèse présentée pour obtenir le
Page 3:
Abstract This thesis deals with the
Page 7 and 8:
Contents Remarks on notation 11 Int
Page 9:
CONTENTS 9 II Multidimensional mode
Page 12 and 13:
12 For {P n } n≥1 , P ∈ P(Ω) t
Page 14 and 15:
14 INTRODUCTION EN FRANCAIS corresp
Page 16 and 17:
16 INTRODUCTION EN FRANCAIS Lévy p
Page 18 and 19:
18 INTRODUCTION EN FRANCAIS Calibra
Page 20 and 21:
20 INTRODUCTION EN FRANCAIS • Si
Page 22 and 23:
22 INTRODUCTION EN FRANCAIS et Scai
Page 24 and 25:
24 INTRODUCTION EN FRANCAIS Ce rés
Page 26 and 27:
26 INTRODUCTION EN FRANCAIS dans un
Page 28 and 29:
28 INTRODUCTION 7500 Taux de change
Page 30 and 31:
30 INTRODUCTION Lévy models by Fou
Page 33 and 34:
Chapter 1 Lévy processes and exp-L
Page 35 and 36:
1.1. LEVY PROCESSES 35 Proposition
Page 37 and 38:
1.1. LEVY PROCESSES 37 The characte
Page 39 and 40:
1.1. LEVY PROCESSES 39 On the other
Page 41 and 42:
1.2. EXPONENTIAL LEVY MODELS 41 (b)
Page 43 and 44:
1.3. EXP-LEVY MODELS IN FINANCE 43
Page 45 and 46:
1.3. EXP-LEVY MODELS IN FINANCE 45
Page 47 and 48:
1.4. PRICING EUROPEAN OPTIONS 47 wi
Page 49 and 50:
1.4. PRICING EUROPEAN OPTIONS 49 Pr
Page 51 and 52:
1.4. PRICING EUROPEAN OPTIONS 51 Nu
Page 53 and 54:
1.4. PRICING EUROPEAN OPTIONS 53 By
Page 55 and 56:
1.4. PRICING EUROPEAN OPTIONS 55 Si
Page 57 and 58:
Chapter 2 The calibration problem a
Page 59 and 60:
2.1. LEAST SQUARES CALIBRATION 59 m
Page 61 and 62:
2.1. LEAST SQUARES CALIBRATION 61 i
Page 63 and 64:
2.1. LEAST SQUARES CALIBRATION 63 w
Page 65 and 66:
2.1. LEAST SQUARES CALIBRATION 65 A
Page 67 and 68:
2.1. LEAST SQUARES CALIBRATION 67 i
Page 69 and 70:
2.2. SELECTION USING RELATIVE ENTRO
Page 71 and 72:
2.2. SELECTION USING RELATIVE ENTRO
Page 73 and 74:
2.3. RELATIVE ENTROPY IN THE LITERA
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
2.4. RELATIVE ENTROPY OF LEVY PROCE
Page 83 and 84:
2.4. RELATIVE ENTROPY OF LEVY PROCE
Page 85 and 86:
2.5. REGULARIZING THE CALIBRATION P
Page 87 and 88:
2.5. REGULARIZING THE CALIBRATION P
Page 89 and 90: 2.5. REGULARIZING THE CALIBRATION P
Page 95 and 96: Chapter 3 Numerical implementation
Page 97 and 98: 3.1. DISCRETIZING THE CALIBRATION P
Page 99 and 100: 3.2. CHOICE OF THE PRIOR 99 Since
Page 101 and 102: 3.2. CHOICE OF THE PRIOR 101 twice
Page 103 and 104: 3.3. CHOICE OF THE REGULARIZATION P
Page 111 and 112: 3.4. CHOICE OF THE WEIGHTS 111 Then
Page 113 and 114: 3.5. NUMERICAL SOLUTION 113 have th
Page 115 and 116: 3.5. NUMERICAL SOLUTION 115 of X t
Page 117 and 118: 3.5. NUMERICAL SOLUTION 117 since
Page 119 and 120: 3.6. NUMERICAL AND EMPIRICAL TESTS
Page 131: Part II Multidimensional modelling
Page 134 and 135: 134 CHAPTER 4. DEPENDENCE OF LEVY P
Page 166 and 167: 166 CHAPTER 5. APPLICATIONS OF LEVY
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
Page 198 and 199:
198 BIBLIOGRAPHY
Page 200:
200 INDEX tightness, 40 levycalibra
show all

Processus de LÃ©vy en Finance - Laboratoire de ProbabilitÃ©s et ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?