On the Robust Dynkin Game

Abstract

We analyze a robust version of the Dynkin game over a set P 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. If the probability set P is weakly compact, one can even find an optimal triplet for the value V0. The mutual singularity of probabilities in P causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times.