Stability of the Exit Time for Lévy Processes

Abstract

This paper is concerned with the behaviour of a Lévy process when it crosses over a positive level, $u$, starting from 0, both as $u$ becomes large and as $u$ becomes small. Our main focus is on the time, ${\tau }_u$, it takes the process to transit above the level, and in particular, on the {\it stability} of this passage time; thus, essentially, whether or not ${\tau }_u$ behaves linearly as $u\dto 0$ or $u\to \infty$. We also consider conditional stability of ${\tau }_u$ when the process drifts to $-\infty$, a.s. This provides information relevant to quantities associated with the ruin of an insurance risk process, which we analyse under a Cramér condition.