Generalized Dynkin games and doubly reflected BSDEs with jumps

Abstract

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.

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