We study the existence of optimal actions in a zero-sum game \inf _\tau \sup_PE ^P[X_\tau] between a stopper and a controller choosing a probability measure. In particular, we consider the optimal stopping problem \inf _\tau E(X _\tau ) for a class of sublinear expectations 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.
We study the existence of optimal actions in a zero-sum game \inf _{\tau \sup_PE ^P[X_{\tau between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem \inf _{\tau E(X _{\tau ) for a class of sublinear expectations 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.
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