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

5.1. PARAMETRIC FAMILIES 173
Lévy copulas (one for each orthant) as follows:
Theorem 5.3. For each {α 1 , . . . , α d } ∈ {−1, 1} d let g (α 1,...,α d ) (u) : [0, ∞] → [0, 1] be a nonnegative,
increasing function satisfying
∑
α∈{−1,1} d with α k =−1
g (α 1,...,α d ) (u) = 1
and
∑
α∈{−1,1} d with α k =1
g (α 1,...,α d ) (u) = 1
for all u ∈ [0, ∞] and all k ∈ {1, . . . , d}. Moreover, let F (α 1,...,α d ) be a positive Lévy copula
that satisfies the following continuity property at infinity: for all I ⊂ {k : α k = −1}, (u i ) i∈I c ∈
[0, ∞] Ic we have
lim F (α 1,...,α d ) (u 1 , . . . , u d ) = F (α 1,...,α d ) (v 1 , . . . , v d ),
{u i } i∈I →(∞,...,∞)
where v i = u i for i ∈ I c and v i = ∞ otherwise. Then
F (u 1 , . . . , u d ) :=
( ) d∏
F (sgn u 1,...,sgn u d )
|u 1 |g (sgn u 1,...,sgn u d ) (|u 1 |), . . . , |u d |g (sgn u 1,...,sgn u d ) (|u d |) sgn u i
defines a Lévy copula.
i=1
Proof. Properties 1 and 2 of Definition 4.16 are obvious. Property 3 follows after observing that
u ↦→ ug (α 1,...,α d ) (u) is increasing on [0, ∞] for any {α 1 , . . . , α d } ∈ {−1, 1} d . To prove property
4, note that
( )
F (α 1,...,α d )
|u 1 |g (α 1,...,α d ) (|u 1 |), . . . , |u d |g (α 1,...,α d ) (|u d |) = |u k |g (α 1,...,α d ) (|u k |)
for any {α 1 , . . . , α d } ∈ {−1, 1} d and any {u 1 , . . . , u d } ∈ ¯R d with u i = ∞ for all i ≠ k. Therefore,
⎧
⎪⎨
F {k} (u) =
⎪⎩
∑
d∏ ∏
ug (α 1,...,α d ) (u) α i α j if u ≥ 0
α∈{−1,1} d with α k =1
∑
i=1
|u|g (α 1,...,α d ) (|u|)
j≠k
α∈{−1,1} d with α k =−1
i=1 j≠k
d∏ ∏
α i α j if u < 0
= u.