## The Annals of Applied Probability

### 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.

#### Article information

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

Dates
Revised: June 2014
First available in Project Euclid: 30 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1438261047

Digital Object Identifier
doi:10.1214/14-AAP1054

Mathematical Reviews number (MathSciNet)
MR3375882

Zentralblatt MATH identifier
1322.60047

#### Citation

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. https://projecteuclid.org/euclid.aoap/1438261047

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