Optimal Stopping under Adverse Nonlinear Expectation and Related Games

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. In particular, we consider the optimal stopping problem $\inf_\tau \mathcal{E}(X_\tau)$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ including 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.