Reflected BSDEs and mixed game problem

Under smooth assumptions and notably the Isaacsʹs condition on the Hamiltonian, we prove the existence of a saddle-point for the “mixed” zero-sum stochastic differential game with payoffJ(u,τ;v,σ)=E(u, v)∫0τ∧σf(s,X,us,vs) ds+Lτ1[τ⩽σ, σ<T]+Uσ1[σ<τ]+ξ1[τ∧σ=T].The main tool is the notion of double barrier reflected backward SDEs.