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

138 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES
Since C Z (u, v) is different from C ⊥ (u, v), we conclude that C t is not constant over time.
Every Lévy process X := {X t } t≥0 is described in a time-dependent fashion by its characteristic
triplet (A, ν, γ). It seems therefore natural to describe the dependence between components
of X also in terms of its characteristic triplet. Since the continuous martingale component of
X is completely described by the covariance matrix A and is independent from the jump part,
we will focus on the dependence of the jump part of X, that is, we only consider Lévy processes
with A = 0. For such a process, separate modelling of margins and dependence will be
achieved by introducing Lévy copulas, which play the same role for Lévy measures as copulas
for probability measures. After showing how basic dependence patterns for Lévy processes are
expressed in terms of their characteristic triplets in Section 4.2 and introducing the necessary
notation and definitions in Section 4.3, first, in Section 4.4 we treat the conceptually simpler
case of Lévy processes, admitting only positive jumps in every component or, equivalently, having
Lévy measures supported by [0, ∞) d . Lévy copulas for general Lévy processes, introduced
in a joint work of the present author with Jan Kallsen [59], are discussed in Sections 4.5 and
4.6 of this chapter.
4.2 Dependence concepts for multidimensional Lévy processes
In this section, X := {X i t} i=1,...,d
t≥0
(A, ν, γ).
denotes a Lévy process on R d with characteristic triplet
For I ⊂ {1, . . . , d} we define I c := {1, . . . , d} \ I and |I| denotes the number of
elements in I. We start by defining the margins of a Lévy process.
Definition 4.1. Let I ⊂ {1, . . . , d} nonempty. The I-margin of X is the Lévy process X I :=
{X i t} i∈I
t≥0 .
The following lemma explains that the Lévy measure of X I
measure of X and shows how it can be computed.
only depends on the Lévy
Lemma 4.1 (Marginal Lévy measures). Let I ⊂ {1, . . . , d} nonempty.
process X I has Lévy measure ν I given by
Then the Lévy
ν I (B) = ν({x ∈ R d : (x i ) i∈I ∈ B}), ∀B ∈ B(R |I| \ {0}). (4.1)
Proof. This lemma is a direct consequence of Proposition 11.10 in [87].