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 {M_t} for processes of the form Z_t=X_t+Y_t where {X_t} is a Lévy process and Y_t satisfies certain regularity conditions. In particular, this provides a martingale for the case where Y_t=L_t where L_t is the local time at zero of the corresponding reflected Lévy process. In this case {M_t} involves, among others, the Lévy exponent φ(α) and L_t. In this paper, conditions for optional stopping of {M_t} 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 {e^{αX>t-tφ(α)}}, namely that P̃(τ<∞)=1 where P̃ corresponds to a suitable exponentially tilted Lévy process.