Generalized Dynkin Games and Doubly Reflected BSDEs with Jumps

Abstract

We introduce a generalized Dynkin game problem with non linear conditional expectation $\mathcal{E}$ induced by a Backward Stochastic Differential Equation (BSDE) with jumps. Let $\xi ,\zeta$ be two RCLL adapted processes with $\xi \le \zeta$. The criterium is given by \begin{equation*} {\cal J}_{τ, σ}= {\cal E}_{0, τ\wedge σ} \left(ξ_τ\textbf{1}_{\{ τ\leq σ\}}+ζ_σ\textbf{1}_{\{σ<τ\}}\right) \end{equation*} where $\tau$ and $\sigma$ are stopping times valued in $[0,T]$. Under Mokobodski's condition, we establish the existence of a value function for this game, i.e. ${\inf }_{\sigma }{\sup }_{\tau }{\mathcal{J}}_{\tau ,\sigma }={\sup }_{\tau }{\inf }_{\sigma }{\mathcal{J}}_{\tau ,\sigma }$. 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 address the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.

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