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

4.4. LEVY COPULAS: SPECTRALLY POSITIVE CASE 147
Lévy copulas in the spectrally positive case, including a characterization of the convergence of
Lévy measures in terms of Lévy copulas can be found in a recent paper by Barndorff-Nielsen
and Lindner [8].
As the laws of random variables are represented by their distribution functions, Lévy measures
can be represented by their tail integrals.
Definition 4.12. Let ν be a Lévy measure on R d + := [0, ∞) d \ {0}. The tail integral U of ν is
a function [0, ∞) d → [0, ∞] such that
1. U(0, . . . , 0) = ∞.
2. For (x 1 , . . . , x d ) ∈ R d +,
U(x 1 , . . . , x d ) = ν([x 1 , ∞) × · · · × [x d , ∞)).
The Lévy measure is uniquely determined by its tail integral, because the above definition
implies that for every x, y ∈ R d + with x ≤ y,
V U (|x, y|) = (−1) d ν([x 1 , y 1 ) × · · · × [x d , y d )). (4.14)
Definition 4.13. Let X be a R d -valued Lévy process and let I ⊂ {1, . . . , d} non-empty. The
I-marginal tail integral U I of X is the tail integral of the process X I := (X i ) i∈I . The onedimensional
margins are, as usual, denoted by U i := U {i} .
Lemma 4.1 entails that for I ⊂ {1, . . . , d} nonempty, the I-marginal tail integral of a Lévy
measure ν on R d + can be computed from the tail integral U of ν by substituting 0 instead of
arguments with indices not in I:
U I ((x i ) i∈I ) = U(x 1 , . . . , x d )| (xi ) i∈I c =0. (4.15)
A Lévy copula is defined similarly to ordinary copula but on a different domain.
Definition 4.14. A function F : [0, ∞] d → [0, ∞] is a Lévy copula if
1. F (u 1 , . . . , u d ) < ∞ for (u 1 , . . . , u d ) ≠ (∞, . . . , ∞),
2. F is grounded: F (u 1 , . . . , u d ) = 0 whenever u i = 0 for at least one i ∈ {1, . . . , d},