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

4.2. DEPENDENCE CONCEPTS FOR LEVY PROCESSES 139
In view of the above lemma, for a given Lévy measure ν we will refer to the Lévy measure
ν I defined by Equation (4.5) as the I-margin of ν. To simplify notation, when I = {k} for some
k, the I-margin of ν will be denoted by ν k and called simply k-th margin of ν.
Next we would like to characterize the independence of Lévy processes in terms of their
characteristic triplets.
Lemma 4.2. The components X 1 , . . . , X d of an R d -valued Lévy process X are independent
if and only if their continuous martingale parts are independent and the Lévy measure ν is
supported by the coordinate axes. ν is then given by
ν(B) =
d∑
ν i (B i ) ∀B ∈ B(R d \ {0}), (4.2)
i=1
where for every i, ν i denotes the i-th margin of ν and
B i = {x ∈ R : ( 0, . . . , 0
} {{ }
, x, 0, . . . , 0) ∈ B}.
i − 1 times
Proof. Since the continuous martingale part and the jump part of X are independent, we
can assume without loss of generality that X has no continuous martingale part, that is, its
characteristic triplet is given by (0, ν, γ).
The “if” part. Suppose ν is supported by the coordinate axes. Then necessarily for every
B ∈ B(R d \ {0}), ν(B) = ∑ d
i=1 ˜ν i(B i ) with some measures ˜ν i , and Lemma 4.1 show that these
measures coincide with the margins of ν: ˜ν i = ν i ∀i. Using the Lévy-Khintchine formula for
the process X, we obtain:
∫
E[e i〈u,Xt〉 ] = exp t{i〈γ, u〉 + (e i〈u,x〉 − 1 − i〈u, x〉1 |x|≤1 )ν(dx)}
R d \{0}
d∑
∫
= exp t {iγ k u k + (e iu kx k
− 1 − iu k x k 1 |xk |≤1)ν k (dx k )} =
k=1
R\{0}
which shows that the components of X are independent Lévy processes.
d∏
E[e iu kXt k ],
The “only if” part. Define a measure ˜ν on R d \ {0} by ˜ν(B) = ∑ d
i=1 ν i(B i ), where ν i is the
i-th marginal Lévy measure of X and B i is as above. It is straightforward to check that ˜ν is
a Lévy measure. Since the components of X are independent, applying the Lévy-Khintchine
formula to each component of X, we find:
∫
E[e i〈u,Xt〉 ] = exp t{i〈γ, u〉 +
R d \{0}
k=1
(e i〈u,x〉 − 1 − i〈u, x〉1 |x|≤1 )˜ν(dx)}.