A Distributional Equality for Suprema of Spectrally Positive Lévy Processes

Abstract

Let $$Y$$ be a spectrally positive Levy process with $${\mathbb {E}}Y_1\!<\!0$$ and $$C$$ an independent subordinator with finite expectation, and let $$X\!=\!Y\!+\!C$$ . A curious distributional equality proved in Huzak et al. (Ann Appl Probab 14:1278–1397, 2004) states that if $${\mathbb {E}}X_1<0$$ , 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.