On the Robust Dynkin Game

We study a robust Dynkin game over a set of mutually singular probabilities. We first prove that for the conservative player of the game, her lower and upper value processes coincide (i.e. She has a value process $V$ in the game). Such a result helps people connect 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 expectations up to the first time $\tau_*$ when $V$ meets the lower payoff process $L$. If the probability set is weakly compact, one can even find an optimal triplet. 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. The mutual singularity of probabilities causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times