First exit times from a bounded interval for L\'{e}vy processes

TL;DR

This paper provides sharp estimates for the mean first exit time of Lévy processes from bounded intervals, analyzes the distribution of extrema, and establishes conditions for ladder height renewal functions.

Contribution

It introduces new sharp two-sided estimates for the mean exit time and integral conditions for ladder height processes, advancing understanding of Lévy process boundary behaviors.

Findings

Sharp two-sided estimates for mean exit times

Distributional analysis of supremum and infimum processes

Integral conditions for ladder height renewal functions

Abstract

In this paper we study the mean of the first exit time from a bounded interval of various L\'evy processes. We establish sharp two-sided estimates of the mean for L\'evy processes under certain condition on their characteristic exponents. We also study the cumulative distribution function of the supremum and infimum processes. Finally, we establish integral conditions that assure that the renewal function of the ladder height process is comparable with the linear one.