Convolution equivalent L\'{e}vy processes and first passage times

TL;DR

This paper analyzes the behavior of convolution equivalent Lévy processes at first passage times, providing asymptotic estimates and limiting process characterizations relevant for insurance risk modeling.

Contribution

It offers new asymptotic estimates for first passage probabilities and demonstrates the existence of a limiting process as the threshold goes to infinity.

Findings

Precise asymptotic probability estimates for first passage times

Existence of a limiting process as the threshold increases

Enhanced understanding of overshoot behavior in Lévy processes

Abstract

We investigate the behavior of L\'{e}vy processes with convolution equivalent L\'{e}vy 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.