Abstract

The Wiener-Hopf factorization is a fundamental result in the theory of Levy processes; it provides a wealth of information about the first exit of the underlying process from a half-line. The main goal of these lectures is to show how to use complex-analytic methods to obtain explicit formulas for Wiener-Hopf factors for several important classes of Levy processes. We will start with processes with jumps of rational transform, then we will discuss the class of stable processes, explaining how one could recover from the Wiener-Hopf factors the distribution of the supremum of the process at a fixed time. Finally, we will talk about the difficult problem of how a Levy process exits an interval, which turns out to be related to Wiener-Hopf factorization for certain 2×2 matrices. This latter problem is wide open for processes with double-sided jumps and we will discuss what is currently known for stable processes.