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

5.2. SIMULATION OF DEPENDENT LEVY PROCESSES 179
Moreover, for every f ∈ C b (R d ) such that f(x) ≡ 0 on a neighborhood of 0,
∫
∫
f(x)ν τ (dx) = f(x)ν(dx)
R d R d
starting from a sufficiently large τ. Therefore, Proposition 1.7 allows to conclude that {Z τ,t } 0≤t≤1
converges in law to a Lévy process with characteristic function given by (5.13).
If the Lévy process has paths of infinite variation on compacts, it can no longer be represented
as the sum of its jumps and we have to introduce a centering term into the series
(5.12).
Theorem 5.7. (Simulation of multidimensional Lévy processes, infinite variation
case)
Let ν be a Lévy measure on R d with marginal tail integrals U i , i = 1, . . . , d and Lévy copula
F (x 1 , . . . , x d ), such that the condition (5.5) is satisfied. Let {V i } and {Γ 1 i }, . . . , {Γd i } be as in
Theorem 5.6. Let
∫
A k (τ) =
|x|≤1
where ν τ is given by (5.15). Then the process
{Z τ,t } 0≤t≤1 , where Z k τ,t = ∑
x k ν τ (dx 1 · · · dx d ), k = 1 . . . d,
U (−1)
k
−τ≤Γ 1 i ≤τ
(Γ k i )1 Vi ≤t − tA k (τ),
converges in law as τ → ∞ to a Lévy process {Z t } 0≤t≤1 on the time interval [0, 1] with characteristic
function
( ∫
)
e i〈u,Zt〉 = exp t (e i〈u,z〉 − 1 − i〈u, z〉)1 |z|≤1 ν(dz) . (5.16)
R d
Proof. The proof is essentially the same as in Theorem 5.6. Similarly to Equation (5.14), Z k τ,s
can now be represented as
Z k τ,s =
∫
[0,s]×{x∈R d :|x|≤1}
x k {N τ (ds × dx 1 · · · dx d ) − dsν τ (dx 1 · · · dx d )}
∫
+
x k N τ (ds × dx 1 · · · dx d ),
[0,s]×{x∈R d :|x|>1}
where N τ is a Poisson random measure on [0, 1] × R d with intensity measure λ [0,1] (ds) ⊗ ν τ ,
and ν τ is defined by (5.15). This entails that Z τ,s is a Lévy process (compound Poisson) with