On the Robust Optimal Stopping Problem

SIAM Journal on Control and Optimization, Vol. 52, No. 5, pp. 3135–3175, 2013

50 Pages Posted: 2 Feb 2014 Last revised: 10 Jul 2016

Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

Song Yao

University of Pittsburgh

Date Written: December 21, 2013

Abstract

We study a robust optimal stopping problem with respect to a set P 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 P. We show that the upper Snell envelope Z of the reward process Y is a supermartingale with respect to an appropriately defined nonlinear expectation E, and Z is further an E-martingale up to the first time tau  when Z meets Y . Consequently, tau 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.

Keywords: robust optimal stopping, zero-sum game of control and stopping, volatility uncertainty, dynamic programming principle, Snell envelope, nonlinear expectation, weak stability under pasting, path-dependent stochastic differential equations with controls

Suggested Citation

Bayraktar, Erhan and Yao, Song, On the Robust Optimal Stopping Problem (December 21, 2013). SIAM Journal on Control and Optimization, Vol. 52, No. 5, pp. 3135–3175, 2013. Available at SSRN: https://ssrn.com/abstract=2388782 or http://dx.doi.org/10.2139/ssrn.2388782

Erhan Bayraktar (Contact Author)

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Song Yao

University of Pittsburgh ( email )

507 Thackeray Hall
Pittsburgh, PA 15260
United States

Paper statistics

Downloads
33
Abstract Views
278