Suprema of L\'{e}vy processes

TL;DR

This paper investigates the distribution of the supremum of one-dimensional Lévy processes, providing uniform estimates and integral representations of their distribution functions under mild conditions.

Contribution

It offers new uniform estimates for the supremum distribution and an integral representation of its Laplace transform in the symmetric case with increasing Lévy-Khintchin exponent.

Findings

Uniform estimate of the supremum's distribution function

Integral representation of the Laplace transform in symmetric cases

Results apply under very mild assumptions

Abstract

In this paper we study the supremum functional $M_t=\sup_{0\le s\le t}X_s$, where $X_t$, $t\ge0$, is a one-dimensional L\'{e}vy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of $M_t$. In the symmetric case we find an integral representation of the Laplace transform of the distribution of $M_t$ if the L\'{e}vy-Khintchin exponent of the process increases on $(0,\infty)$.