Optimal stopping under adverse nonlinear expectation and related games

We study the existence of optimal actions in a zero-sum game inf.supPEP[X.] between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem inf.E(X.) for a class of sublinear expectations E(.) 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:Yt=Xt} 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.