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

Søren Asmussen*, Offer Kella

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

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 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 {eαX>t-tφ(α)}, namely that P̃(τ<∞)=1 where P̃ corresponds to a suitable exponentially tilted Lévy process.

Original languageAmerican English
Pages (from-to)47-55
Number of pages9
JournalStochastic Processes and their Applications
Volume91
Issue number1
DOIs
StatePublished - Jan 2001

Keywords

  • 60J30
  • Exponential change of measure
  • Local time
  • Lévy process
  • Stopping time
  • Wald martingale

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