Optimal stopping under adverse nonlinear expectation and related games

Abstract

We study the existence of optimal actions in a zero-sum game ${\inf }_{\tau }{\sup }_P{E}^P[X_{\tau }]$ between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem ${\inf }_{\tau }\mathcal{E}(X_{\tau })$ for a class of sublinear expectations $\mathcal{E}(\cdot )$ such as the $G$-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear expectations, we define a nonlinear Snell envelope $Y$ and prove that the first hitting time $\inf \{t:Y_t=X_t\}$ is an optimal stopping time. The existence of a saddle point is shown under a compactness condition. Finally, the results are applied to the subhedging of American options under volatility uncertainty.

Published at http://dx.doi.org/10.1214/14-AAP1054 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Constructing Sublinear Expectations on Path Space
• Martingale approach to stochastic differential games of control and stopping
• Second order reflected backward stochastic differential equations

Cited by in corpus (5)

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• A zero-sum stochastic differential game with impulses, precommitment, and unrestricted cost functions
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• Nash equilibria for game contingent claims with utility-based hedging