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

180 CHAPTER 5. APPLICATIONS OF LEVY COPULAS
1
0.8
0.6
0.4
0.2
−10 −8 −6 −4 −2 0 2 4 6 8 10
Figure 5.3: Conditional distribution function F ξ , corresponding to the Lévy copula of Example
5.1 with ξ = 1, θ = 1 and η = 0.25.
characteristic function
e i〈u,Zτ,t〉 = exp
( ∫
)
t (e i〈u,z〉 − 1 − i〈u, z〉1 |z|≤1 )ν τ (dz) .
R d
Proposition 1.7 once again allows to conclude that (Z τ,s ) 0≤s≤1 converges in law to a Lévy process
with characteristic function (5.16).
Example 5.3. Let d = 2 and F be the Lévy copula of Example 5.1. A straightforward computation
yields:
F ξ =
⎧ (
⎨
⎩ (1 − η) + 1 +
∣ ) ⎫
ξ ∣∣∣
θ
−1−1/θ
⎬
∣
(η − 1 x2