Continuous-time dynkin games with mixed strategies

Let (X, Y, Z) be a triple of payoff processes defining a Dynkin game R̃(σ,τ) = E[X_σ1_{τ>σ + Y_τ1_{τ<σ} + Z_τ1 _τ=σ],where σ and τ are stopping times valued in [0, T]. In the case Z = Y, it is well known that the condition X ≤ Y is needed in order to establish the existence of value for the game, i.e., inf _τ sup_σ R̃(σ τ) = sup _σ inf_τR̃(σ, τ). In order to remove the condition X ≤ Y, we introduce an extension of the Dynkin game by allowing for an extended set of strategies, namely, the set of mixed strategies. The main result of the paper is that the extended Dynkin game has a value when Z ≤ Y, and the processes X and Y are restricted to be semimartingales continuous at the terminal time T.