On suprema of Lévy processes and application in risk theory

Abstract

Let $\wh{; ; ; X}; ; ; =C-Y$ where $Y$ is a general one-dimensional Levy process and $C$ an independent subordinator. Consider the times when a new supremum of $\wh{; ; ; X}; ; ; $ is reached by a jump of the subordinator $C$. We give a necessary and sufficient condition in order for such times to be discrete. When this is the case and $\wh{; ; ; X}; ; ; $ drifts to $-\infty$, we decompose the absolute supremum of $\wh{; ; ; X}; ; ; $ at these times, and derive a Pollaczek-Hinchin-type formula for the distribution function of the supremum.