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

4.6. EXAMPLES OF LEVY COPULAS 159
X. Therefore, ν is the Lévy measure of X (cf. Lemma 4.7), which entails by Lemma 4.2 that
X 1 , . . . , X d are independent.
Theorem 4.10. The components X 1 , . . . , X d of an R d -valued Lévy process X are independent
if and only if their Brownian motion parts are independent and X has a Lévy copula of the
form
F ⊥ (x 1 , . . . , x d ) :=
d∑ ∏
x i 1 {∞} (x j ) (4.26)
i=1 j≠i
Proof. It is straightforward to see that Equation (4.26) defines a Lévy copula.
The “if” part. Let I ⊂ {1, . . . , d} with card I ≥ 2. Equation (4.23) entails that F⊥ I ((u i) i∈I ) =
0 for all (u i ) i∈I ∈ R I . Therefore, the tail integrals of X satisfy U I ((x i ) i∈I ) = 0 for all (x i ) i∈I ∈
(R \ {0}) I by (4.24). From Lemma 4.9 we conclude that X 1 , . . . , X d are independent.
The “only if” part. By Lemma 4.9, the tail integrals of X satisfy U I ((x i ) i∈I ) = 0 for all
(x i ) i∈I ∈ (R \ {0}) I and all I ⊂ {1, . . . , d} with card I ≥ 2. Since also F⊥ I ((u i) i∈I ) = 0, we
conclude that F ⊥ is a Lévy copula for X.
Observing that F ⊥ | [0,∞] d
following easy corollary.
is a Lévy copula in the sense of Definition 4.14, we obtain the
Corollary 4.1. The components X 1 , . . . , X d of an R d -valued spectrally positive Lévy process X
are independent if and only if their Brownian motion parts are independent and X has a Lévy
copula on [0, ∞] d of the form F ⊥ | [0,∞] d, where F ⊥ is defined by (4.26).
The following theorem describes complete jump dependence in terms of Lévy copulas.
Theorem 4.11. Let X be a R d -valued Lévy process whose Lévy measure is supported by an
ordered set S ⊂ K, where K is as in Equation (4.3). Then the complete dependence Lévy
copula given by
d∏
F ‖ (x 1 , . . . , x d ) := min(|x 1 |, . . . , |x d |)1 K (x 1 , . . . , x d ) sgn x i (4.27)
is a Lévy copula of X.
Conversely, if F ‖ is a Lévy copula of X, then the Lévy measure of X is supported by an
ordered subset of K. If, in addition, the tail integrals U i of X i are continuous and satisfy
lim x→0 U i (x) = ∞, i = 1, . . . , d, then the jumps of X are completely dependent.
i=1