On the robust dynkin game

We analyze a robust version of the Dynkin game over a set .. of mutually singular probabilities. We first prove that conservative player's lower and upper value coincide (let us denote the value by V). Such a result connects the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process V is a submartingale under an appropriately defined nonlinear expectation ... up to the first time .* when V meets the lower payoff process L. If the probability set .. is weakly compact, one can even find an optimal triplet (.*, .*, .*) for the value V0. The mutual singularity of probabilities in .. causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times.