Suprema of Lévy processes

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 Lévy 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 Lévy–Khintchin exponent of the process increases on $(0,\infty)$.