On optional stopping of some exponential martingales for Lévy processes with or without reflection

Kella and Whitt (J. Appl. Probab. 29 (1992) 396) introduced a martingale {Mt} for processes of the form Zt=Xt+Yt where {Xt} is a Lévy process and Yt satisfies certain regularity conditions. In particular, this provides a martingale for the case where Yt=Lt where Lt is the local time at zero of the corresponding reflected Lévy process. In this case {Mt} involves, among others, the Lévy exponent ϕ(α) and Lt. In this paper, conditions for optional stopping of {Mt} at τ are given. The conditions depend on the signs of α and ϕ(α). In some cases optional stopping is always permissible. In others, the conditions involve the well-known necessary and sufficient condition for optional stopping of the Wald martingale {eαXt−tϕ(α)}, namely that P̃(τ<∞)=1 where P̃ corresponds to a suitable exponentially tilted Lévy process.