Abstract (EN)

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 Eg, 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 Jτ,σ=Eg0,τ∧σ(ξτ1{τ≤σ}+ζσ1{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. infσsupτJτ,σ=supτinfσJτ,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.