Exponential moments of first passage times and related quantities for L\'evy processes

TL;DR

This paper establishes criteria for the finiteness of exponential moments of key passage-related times in Lévy processes and analyzes their asymptotic behavior as the threshold tends to infinity.

Contribution

It provides comprehensive criteria and asymptotic results for exponential moments of passage times in Lévy processes, advancing understanding of their probabilistic properties.

Findings

Criteria for finiteness of exponential moments of passage times

Asymptotic behavior of these moments as thresholds grow large

Complete characterization for Lévy processes

Abstract

For a L\'evy process on the real line, we provide complete criteria for the finiteness of exponential moments of the first passage time into the interval $(r,\infty)$, the sojourn time in the interval $(-\infty,r]$, and the last exit time from $(-\infty,r]$. Moreover, whenever these quantities are finite, we derive their respective asymptotic behavior as $r \to \infty$.