On the scaling property in fluctuation theory for stable Lévy processes

Abstract

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 $\{X_{T_{[x,+\infty [}}-x\le h\}$ 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 $X$.

References in corpus (2)

• Some remarks on first passage of Levy processes, the American put and pasting principles
• Overshoots and undershoots of Lévy processes