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

1.1. LEVY PROCESSES 35
Proposition 1.2 (Moments and cumulants of a Lévy process).
1. Let {X t } t≥0 be a Lévy process on R with characteristic triplet (A, ν, γ) and let n ≥ 1.
E[|X t | n ] < ∞ for some t > 0 or equivalently for every t if and only if ∫ |x|≥1 |x|n ν(dx) < ∞.
In this case Φ t (z), the characteristic function of X t , is of class C n and the first n moments
of X t can be computed by differentiation:
E[Xt k ] = 1 ∂ k
i k ∂z k Φ t(z)| z=0 , k = 1, . . . , n.
The cumulants of X t , defined by
c k (X t ) := 1 ∂ k
i k ∂z k log Φ t(z)| z=0 ,
have a particularly simple structure:
∫
c 1 (X t ) ≡ E[X t ] = t(γ + xν(dx)),
c 2 (X t ) ≡ Var X t = t(A +
c k (X t ) = t
∫ ∞
−∞
|x|≥1
∫ ∞
−∞
x 2 ν(dx)),
x k ν(dx) for 3 ≤ k ≤ n.
2. Let {X t } t≥0 be a Lévy process on R with characteristic triplet (A, ν, γ) and let u ∈ R.
E[e uXt ] < ∞ for some t or, equivalently, for all t > 0 if and only if ∫ |x|≥1 eux ν(dx) < ∞.
In this case
E[e uXt ] = e tψ(−iu) .
where ψ is the characteristic exponent of the Lévy process defined by (1.1).
Corollary 1.1. Let {X t } t≥0 be a Lévy process on R with characteristic triplet (A, ν, γ).
1. {X t } t≥0 is a martingale if and only if ∫ |x|≥1
|x|ν(dx) < ∞ and
∫
γ + xν(dx) = 0.
|x|≥1
2. {exp(X t )} t≥0 is a martingale if and only if ∫ |x|≥1 ex ν(dx) < ∞ and
∫
A
∞
2 + γ + (e x − 1 − x1 |x|≤1 )ν(dx) = 0. (1.2)
−∞