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

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  • Asmussen, Søren
  • Kella, Offer

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.

Suggested Citation

  • Asmussen, Søren & Kella, Offer, 2001. "On optional stopping of some exponential martingales for Lévy processes with or without reflection," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 47-55, January.
    • Handle: RePEc:eee:spapps:v:91:y:2001:i:1:p:47-55
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    Cited by:

    1. Asmussen, Søren & Fuckerieder, Pascal & Jobmann, Manfred & Schwefel, Hans-Peter, 2002. "Large deviations and fast simulation in the presence of boundaries," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 1-23, November.

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