Convolution equivalent Lévy processes and first passage times

Abstract

We investigate the behavior of Lévy processes with convolution equivalent Lévy measures, up to the time of first passage over a high level $u$. Such problems arise naturally in the context of insurance risk where $u$ is the initial reserve. We obtain a precise asymptotic estimate on the probability of first passage occurring by time $T$. This result is then used to study the process conditioned on first passage by time $T$. The existence of a limiting process as $u\to\infty$ is demonstrated, which leads to precise estimates for the probability of other events relating to first passage, such as the overshoot. A discussion of these results, as they relate to insurance risk, is also given.