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

158 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES
for any non-empty I ⊂ {1, . . . , d} and any (x i ) i∈I ∈ (R \ {0}) I . Suppose for ease of notation
that x i > 0, i ∈ I. Then
U I ν ((x i ) i∈I ) = ν({ξ ∈ R d : ξ i ∈ [x i , ∞), i ∈ I})
= µ({u ∈ (−∞, ∞] d : U (−1)
i
(u i ) ∈ [x i , ∞), i ∈ I})
= µ({u ∈ (−∞, ∞] d : 0 < u i ≤ U(x i ), i ∈ I}).
= F I ((U i (x i )) i∈I ).
This proves in particular that the one-dimensional marginal tail integrals of ν equal U 1 , . . . , U d .
Since the marginals ν i of ν are Lévy measures on R, we have ∫ (x 2 i ∧ 1)ν i(dx i ) < ∞ for
i = 1, . . . , d. This implies
∫
∫
(|x| 2 ∧ 1)ν(dx) ≤
d∑
d∑
∫
(x 2 i ∧ 1)ν(dx)
i=1
i=1
(x 2 i ∧ 1)ν i (dx i ) < ∞
and hence ν is a Lévy measure on R d . The uniqueness of ν follows from the fact that it is
uniquely determined by its marginal tail integrals (cf. Lemma 4.7).
4.6 Examples of Lévy copulas
In this section we derive the form of Lévy copulas corresponding to special dependence structures
of Lévy processes: independence, complete dependence and the dependence of stable
processes. Examples of parametric families of Lévy copulas will be given in the next chapter.
To characterize independence of components of a multidimensional Lévy process in terms of its
Lévy copula, we need to restate Lemma 4.2 in terms of tail integrals.
Lemma 4.9. The components X 1 , . . . , X d of an R d -valued Lévy process X are independent if
and only if their continuous martingale parts are independent and the tail integrals of the Lévy
measure satisfy U I ((x i ) i∈I ) = 0 for all I ⊂ {1, . . . , d} with card I ≥ 2 and all (x i ) i∈I ∈ (R\{0}) I .
Proof. The “only if” part. Let I ⊂ {1, . . . , d} with card I ≥ 2 and (x i ) i∈I ∈ (R \ {0}) I . Then
the components of the Lévy process (X i ) i∈I are independent as well. Applying Lemma 4.2 to
this process, we conclude, using Equation (4.2), that U I ((x i ) i∈I ) = 0.
The “if” part. Let ν be defined by Equation (4.2), where ν i is the Lévy measure of X i for
i = 1, . . . , d. Then all marginal tail integrals of ν coincide with those of the Lévy measure of