Useful martingales for stochastic storage processes with L\'{e}vy-type input

Abstract

In this paper we generalize the martingale of Kella and Whitt to the setting of L\'{e}vy-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 a.s. and in $L^2$. The reflected L\'{e}vy-type process is considered as an example.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1112.475