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.
"Useful martingales for stochastic storage processes with Lévy-type input." J. Appl. Probab. 50 (2) 439 - 449, June 2013. https://doi.org/10.1239/jap/1371648952