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

Søren Asmussen and Offer Kella

Stochastic Processes and their Applications, 2001, vol. 91, issue 1, 47-55

Abstract: 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 [phi]([alpha]) and Lt. In this paper, conditions for optional stopping of {Mt} at [tau] are given. The conditions depend on the signs of [alpha] and [phi]([alpha]). 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[alpha]Xt-t[phi]([alpha])}, namely that where corresponds to a suitable exponentially tilted Lévy process.

Keywords: Exponential; change; of; measure; Lévy; process; Local; time; Stopping; time; Wald; martingale (search for similar items in EconPapers)
Date: 2001
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Citations: View citations in EconPapers (1)

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