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On the Robust Optimal Stopping Problem

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  • Erhan Bayraktar
  • Song Yao

Abstract

We study a robust optimal stopping problem with respect to a set $\cP$ of mutually singular probabilities. This can be interpreted as a zero-sum controller-stopper game in which the stopper is trying to maximize its pay-off while an adverse player wants to minimize this payoff by choosing an evaluation criteria from $\cP$. We show that the \emph{upper Snell envelope $\ol{Z}$} of the reward process $Y$ is a supermartingale with respect to an appropriately defined nonlinear expectation $\ul{\sE}$, and $\ol{Z}$ is further an $\ul{\sE}-$martingale up to the first time $\t^*$ when $\ol{Z}$ meets $Y$. Consequently, $\t^*$ is the optimal stopping time for the robust optimal stopping problem and the corresponding zero-sum game has a value. Although the result seems similar to the one obtained in the classical optimal stopping theory, the mutual singularity of probabilities and the game aspect of the problem give rise to major technical hurdles, which we circumvent using some new methods.

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File URL: http://arxiv.org/pdf/1301.0091
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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1301.0091.

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Date of creation: Jan 2013
Date of revision: Apr 2014
Handle: RePEc:arx:papers:1301.0091

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Web page: http://arxiv.org/

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  1. Frank Riedel, 2009. "Optimal Stopping With Multiple Priors," Econometrica, Econometric Society, vol. 77(3), pages 857-908, 05.
  2. Marcel Nutz & Ramon van Handel, 2012. "Constructing Sublinear Expectations on Path Space," Papers 1205.2415, arXiv.org, revised Apr 2013.
  3. Marcel Nutz & Jianfeng Zhang, 2012. "Optimal Stopping under Adverse Nonlinear Expectation and Related Games," Papers 1212.2140, arXiv.org.
  4. Ioannis Karatzas & (*), S. G. Kou, 1998. "Hedging American contingent claims with constrained portfolios," Finance and Stochastics, Springer, vol. 2(3), pages 215-258.
  5. Karandikar, Rajeeva L., 1995. "On pathwise stochastic integration," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 11-18, May.
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