On the scaling property in fluctuation theory for stable L\'evy processes

TL;DR

This paper derives the joint Laplace transform for the first hitting time and position of a Lévy process, especially stable ones, and investigates the asymptotic behavior of the first passage time conditioned on small overshoot.

Contribution

It provides a new explicit formula for the joint distribution of hitting time and position for Lévy processes, extending previous results and analyzing the small overshoot asymptotics.

Findings

Explicit joint Laplace transform for hitting time and position.

Recovery of the law of the process at hitting time for stable Lévy processes.

Asymptotic behavior of first passage time conditioned on small overshoot.

Abstract

We find an expression for the joint Laplace transform of the law of $(T_{[x,+\infty[},X_{T_{[x,+\infty[}})$ for a L\'evy process $X$, where $T_{[x,+\infty[}$ is the first hitting time of $[x,+\infty[$ by $X$. When $X$ is an $\alpha$-stable L\'evy 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 $X$.