Useful martingales for stochastic storage processes with Lévy-type input

Abstract

In this paper we generalize the martingale of Kella and Whitt to the setting of Lévy-type processes and show that the (local) martingales obtained are in fact square-integrable martingales which upon dividing by the time index converge to zero almost surely and in L². The reflected Lévy-type process is considered as an example.