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

170 CHAPTER 5. APPLICATIONS OF LEVY COPULAS
Corollary 5.1 (Archimedean Lévy copulas for spectrally positive processes). Let
ψ : [0, ∞] → [0, ∞] be a strictly decreasing continuous function with ψ(0) = ∞, ψ(∞) = 0,
having derivatives of orders up to d on (0, ∞) and satisfying
for all x ∈ (0, ∞). Then
(−1) d dd ψ(x)
dx d ≥ 0
is a Lévy copula on [0, ∞] d .
d∑
F (u 1 , . . . , u d ) := ψ( ψ −1 (u i )) (5.2)
i=1
Proof. Let
⎧
⎨ ψ(− log u), u ≥ 0
φ(u) =
, u ∈ [−1, 1].
⎩
−ψ(− log(−u)), u < 0
Then φ satisfies the conditions of Theorem 5.2 and therefore
( d∏
)
¯F (u 1 , . . . , u d ) := φ φ −1 (u i /2 d−1 )
i=1
is a Lévy copula on (−∞, ∞] d . This implies that the function ˜F := ¯F | [0,∞] d has properties 1,
2 and 3 of Definition 4.14. However, it is easy to check that ˜F (u 1 , . . . , u d )| ui =∞,i≠k = u k /2 d−1
and therefore
d∑
2 d−1 ˜F (u1 , . . . , u d ) = 2 d−1 ψ( ψ −1 (u i /2 d−1 ))
is a Lévy copula on [0, ∞] d , which means that (5.2) also defines a positive Lévy copula.
i=1
Example 5.1. Let
with θ > 0 and η ∈ (0, 1). Then
φ(x) := η(− log |x|) −1/θ 1 x≥0 − (1 − η)(− log |x|) −1/θ 1 x