Stability of the Exit Time for L\'evy Processes

TL;DR

This paper investigates the stability of the first passage time over a level for Lévy processes as the level varies, with implications for risk and ruin analysis in insurance models.

Contribution

It provides new insights into the asymptotic behavior and stability of crossing times for Lévy processes, especially under drift and Cramér conditions.

Findings

Stability of crossing times as level approaches zero or infinity.

Conditional stability results when the process drifts to -∞.

Applications to ruin probabilities in insurance risk models.

Abstract

This paper is concerned with the behaviour of a L\'{e}vy 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 Cram\'er condition.