Generalized Dynkin games and doubly reflected BSDEs with jumps.

Summary We introduce a game problem which can be seen as a generalization of the classical Dynkin game problem to the case of a nonlinear expectation ${\cal E}^g$, induced by a Backward Stochastic Differential Equation (BSDE) with jumps with nonlinear driver $g$. Let $\xi, \zeta$ be two RCLL adapted processes with $\xi \leq \zeta$. The criterium is given by $ {\cal J}_{\tau, \sigma}= {\cal E}^g_{0, \tau \wedge \sigma } \left(\xi_{\tau}\textbf{1}_{\{ \tau \leq \sigma\}}+\zeta_{\sigma}\textbf{1}_{\{\sigma<\tau\}}\right)$ where $\tau$ and $ \sigma$ are stopping times valued in $[0,T]$. Under Mokobodzki's condition, we establish the existence of a value function for this game, i.e. $\inf_{\sigma}\sup_{\tau} {\cal J}_{\tau, \sigma} = \sup_{\tau} \inf_{\sigma} {\cal J}_{\tau, \sigma}$. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When $\xi$ and $\zeta$ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then study the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.