A distributional equality for suprema of spectrally positive Lévy processes

Abstract

Let $Y$ be a spectrally positive Lévy process with $E{Y}_1<0$, $C$ an independent subordinator with finite expectation, and $X=Y+C$. A curious distributional equality proved in Huzak et al., Ann. Appl. Probab. 14 (2004) 1278--1397, states that if $E{X}_1<0$, then ${\sup }_{0\le t<\infty }{Y}_t$ and the supremum of $X$ just before the first time its new supremum is reached by a jump of $C$ have the same distribution. In this paper we give an alternative proof of an extension of this result and offer an explanation why it is true.

14 pp