Generalized Dynkin games and doubly reflected BSDEs with jumps

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 ε^g, induced by a Backward Stochastic Differential Equation (BSDE) with jumps with nonlinear driver g. Let ξ, ζ be two RCLL adapted processes with ξ ≤ ζ. The criterium is given by J_τ,σ = ε^g ₀,_τ∧σ (ξ_τ1_{τ≤σ} + ζ_σ1_{σ<τ}), where τ and σ 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_σ sup_τ J_τ,σ = sup_τ inf_σ J_τ,σ. 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 ξ and ζ 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.