The Annals of Applied Probability

Optimal stopping under adverse nonlinear expectation and related games

Marcel Nutz and Jianfeng Zhang

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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.

Article information

Ann. Appl. Probab. Volume 25, Number 5 (2015), 2503-2534.

Received: August 2013
Revised: June 2014
First available in Project Euclid: 30 July 2015

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Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control 49L20: Dynamic programming method 91A15: Stochastic games 60G44: Martingales with continuous parameter 91B28

Controller-and-stopper game optimal stopping saddle point nonlinear expectation $G$-expectation


Nutz, Marcel; Zhang, Jianfeng. Optimal stopping under adverse nonlinear expectation and related games. Ann. Appl. Probab. 25 (2015), no. 5, 2503--2534. doi:10.1214/14-AAP1054.

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