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

154 CHAPTER 4. DEPENDENCE OF LEVY PROCESSES
Let a, b ∈ R d such that a i b i ≤ 0 for at most k indices. For ease of notation we suppose that
a i b i ≤ 0 for i = 1, . . . , k. Let I ⊂ {1, . . . , d} non-empty with 0 /∈ ∏ i∈I [a i, b i ). By induction
hypothesis, ν I ( ∏ i∈I [a i, b i )) is uniquely determined if k ∉ I. Suppose that k ∈ I. If b k = 0, then
( ) ⎛
⎞
∏
ν I i , b i ) = lim ν
α↑0
i∈I[a I ⎝ ∏
[a i , b i ) × [a k , α) × ∏
[a i , b i ) ⎠
i∈I,ik
and the right-hand side is uniquely determined by the induction hypothesis. If b k ≠ 0, then
( )
⎛
⎞
∏
ν I i , b i ) = ν
i∈I[a I\{k} ⎝ ∏
[a i , b i ) ⎠
which is uniquely determined as well.
i∈I\{k}
⎛
⎞
− lim ν I ⎝ ∏
[a i , b i ) × [b k , c) × ∏
[a i , b i ) ⎠
c↑∞
i∈I,ik
⎛
⎞
− lim
α↑a k ;c↓−∞ νI ⎝ ∏
[a i , b i ) × [c, a k ) × ∏
[a i , b i ) ⎠ ,
i∈I,ik
Lévy copulas in the general case are defined similarly to the spectrally positive case.
Definition 4.16. A function F : (−∞, ∞] d → (−∞, ∞] is a Lévy copula if
1. F (u 1 , . . . , u d ) < ∞ for (u 1 , . . . , u d ) ≠ (∞, . . . , ∞),
2. F (u 1 , . . . , u d ) = 0 if u i = 0 for at least one i ∈ {1, . . . , d},
3. F is d-increasing,
4. F i (u) = u for any i ∈ {1, . . . , d}, u ∈ (−∞, ∞].
Remark 4.2. Since F is d-increasing, the sup in Equation (4.8) for the margins of F may be
computed by taking b i = ∞ and a i → −∞ for every i ∈ I c . Therefore, Equation (4.8) reduces
to
F I ((x i ) i∈I ) := lim
∑
c→∞
(x j ) j∈I c ∈{−c,∞} |Ic |
F (x 1 , . . . , x d ) ∏
j∈I c sgn x j . (4.23)
When F is a Lévy copula on [0, ∞] d (Definition 4.14), it can be extended to a Lévy copula F ext
on (−∞, ∞] d by taking
⎧
⎪⎨ F (x 1 , . . . , x d ), (x 1 , . . . , x d ) ∈ [0, ∞] d
F ext (x 1 , . . . , x d ) :=
⎪⎩
0 otherwise.