Processus de Lévy en Finance - Laboratoire de Probabilités et ...

148 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES
3. F is d-increasing,
4. F i (u) = u for any i ∈ {1, . . . , d}, u ∈ [0, ∞].
The following theorem is an analog of the well-known Sklar’s theorem for copulas. The
proof is done using multilinear interpolation and is inspired by Sklar’s proof of his theorem in
[90].
Theorem 4.6. Let ν be a Lévy measure on R d + with tail integral U and marginal Lévy measures
ν 1 , . . . , ν d . There exists a Lévy copula F on [0, ∞] d such that
U(x 1 , . . . , x d ) = F (U 1 (x 1 ), . . . , U d (x d )), (x 1 , . . . , x d ) ∈ [0, ∞) d , (4.16)
where U 1 , . . . , U d are tail integrals of ν 1 , . . . , ν d . This Lévy copula is unique on ∏ d
i=1 Ran U i.
Conversely, if F is a Lévy copula on [0, ∞] d and ν 1 , . . . , ν d are Lévy measures on (0, ∞)
with tail integrals U 1 , . . . , U d then Equation (4.16) defines a tail integral of a Lévy measure on
R d + with marginal Lévy measures ν 1 , . . . , ν d .
Remark 4.1. In particular, the Lévy copula F is unique if the marginal Lévy measures ν 1 , . . . , ν d
are infinite and have no atoms, because in this case Ran U k = [0, ∞] for every k.
The first part of this theorem states that all types of dependence of Lévy processes (with
only positive jumps), including complete dependence and independence, can be represented
with Lévy copulas. The second part shows that one can construct multivariate Lévy process
models by specifying separately jump dependence structure and one-dimensional laws for the
components. The laws of components can have very different structure, in particular, it is possible
to construct examples of Lévy processes with some components being compound Poisson
and others having an infinite jump intensity.
Proof of theorem 4.6. For the purposes of this proof, we introduce some auxiliary functions and
measures. First, for every k = 1, . . . , d and every x ∈ [0, ∞] we define
⎧
⎪⎨ U k (x), x ≠ ∞
Ũ k (x) :=
⎪⎩ 0, x = ∞.
Ũ (−1)
k
(t) := sup{x ≥ 0 : Ũk(x) ≥ t}.