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

TL;DR

This paper extends martingale techniques to Le9vy-type stochastic storage processes, demonstrating their integrability and convergence properties, with applications to reflected Le9vy-type processes.

Contribution

It generalizes the Kella-Whitt martingale to Le9vy-type processes and proves their square integrability and convergence.

Findings

Martingales are square integrable and converge to zero a.s. and in L^2.

The approach applies to reflected Le9vy-type processes.

Provides a framework for analyzing stochastic storage with Le9vy inputs.

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.