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 [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.
Year of publication: |
2001
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Authors: | Asmussen, Søren ; Kella, Offer |
Published in: |
Stochastic Processes and their Applications. - Elsevier, ISSN 0304-4149. - Vol. 91.2001, 1, p. 47-55
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Publisher: |
Elsevier |
Keywords: | Exponential change of measure Lévy process Local time Stopping time Wald martingale |
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