# Stability of the overshoot for Lévy processes

Doney, R. A. and Maller, R.A. (2002) Stability of the overshoot for Lévy processes. The Annals of Probability, 30 (1). pp. 188-212. ISSN 0091-1798

## Abstract

We give equivalences for conditions like $X(T(r))/r\rightarrow 1$ and $X(T^{*}(r))/\allowbreak r\rightarrow 1$, where the convergence is in probability or almost sure, both as $r\rightarrow 0$ and $r\rightarrow \infty$, where $X$ is a L\'{e}vy process and $T(r)$ and $T^{*}(r)$ are the first exit times of $X$ out of the strip $\{(t,y):t> 0,|y|\leq r\}$ and half-plane $\{(t,y):t> 0$, $y\leq r\}$, respectively. We also show, using a result of Kesten, that $X(T^{*}(r))/r\rightarrow 1$ a.s.\ as $r\to 0$ is equivalent to $X$ creeping'' across a level.

Item Type: Article Processes with independent increments; exit times; first passage times; local behavior MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes Ms Lucy van Russelt 06 Sep 2006 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/588