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.
Citation
Marcel Nutz. Jianfeng Zhang. "Optimal stopping under adverse nonlinear expectation and related games." Ann. Appl. Probab. 25 (5) 2503 - 2534, October 2015. https://doi.org/10.1214/14-AAP1054
Information