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

5.2. SIMULATION OF DEPENDENT LEVY PROCESSES 175
Proof. We must prove that for each A ∈ B(R d ),
ν(A) = µ({u ∈ R d : f(u) ∈ A}),
but in view of Lemma 4.7, it is sufficient to show that for each I ⊂ {1, . . . , d} nonempty and
for all (x i ) i∈I ∈ (R \ {0}) |I| ,
U I ((x i ) i∈I ) = µ({u ∈ R d : U (−1)
i
(u i ) ∈ I(x i ), i ∈ I}),
where I(x) was defined in (4.21). However, since U i is left-continuous, for every i, U (−1)
i
(u) ∈
I(x) if and only if u ∈ (U i (x) ∧ 0, U i (x) ∨ 0]. Therefore,
µ({u ∈ R d : U (−1)
i
(u i ) ∈ I(x i ), i ∈ I})
= µ({u ∈ R d : u i ∈ (U i (x i ) ∧ 0, U i (x i ) ∨ 0], i ∈ I}) = F I ((U i (x i )) i∈I ),
and an application of Theorem 4.8 completes the proof.
By Theorem 2.28 in [1], there exists a family, indexed by ξ ∈ R, of positive Radon measures
K(ξ, dx 2 · · · dx d ) on R d−1 , such that
ξ ↦→ K(ξ, dx 2 · · · dx d )
is Borel measurable and
µ(dx 1 . . . dx d ) = λ(dx 1 ) ⊗ K(x 1 , dx 2 · · · dx d ). (5.7)
In addition, K(ξ, R d−1 ) = 1 λ-almost everywhere, that is, K(ξ, ∗) is, almost everywhere, a probability
distribution. In the sequel we will call {K(ξ, ∗)} ξ∈R the family of conditional probability
distributions associated to the Lévy copula F .
Let F ξ be the distribution function of the measure K(ξ, ∗):
F ξ (x 2 , . . . , x d ) := K(ξ, (−∞, x 2 ] × · · · × (−∞, x d ]), (x 2 , . . . , x d ) ∈ R d−1 . (5.8)
The following lemma shows that it can be computed in a simple manner from the Lévy copula
F .