On the scaling property in fluctuation theory for stable Lévy processes

Fernando Cordero (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We find an expression for the joint Laplace transform of the law of $(T_{[x,+\infty[},X_{T_{[x,+\infty[}})$ for a Lévy process $X$, where $T_{[x,+\infty[}$ is the first hitting time of $[x,+\infty[$ by $X$. When $X$ is an $\alpha$-stable Lévy process, with $1<\alpha<2$, we show how to recover from this formula the law of $X_{T_{[x,+\infty[}}$; this result was already obtained by D. Ray, in the symmetric case and by N. Bingham, in the case when $X$ is non spectrally negative. Then, we study the behaviour of the time of first passage $T_{[x,+\infty[},$ conditioned to $\{X_{T_{[x,+\infty[}} -x \leq h\}$ when $h$ tends to $0$. This study brings forward an asymptotic variable $T_x^0$, which seems to be related to the absolute continuity of the law of the supremum of $X$.

Keywords: Fluctuation theory; Scaling property; Lévy processes; Stable processes; Overshoots; First passage time

Domains: Probability [math.PR]

Fernando Cordero. On the scaling property in fluctuation theory for stable Lévy processes. 2010. ⟨hal-00505184⟩