Stochastic Recursive Zero-Sum Differential Game and Mixed Zero-Sum Differential Game Problem with Payoff Functional in BDSDES

In this paper, we study stochastic recursive (mixed) zero-sum differential game and this game problem' payoff functional is depicted by the forms of a backward doubly stochastic differential equation'solution. The major means are backward doubly stochastic differential equations (BDSDEs) and double-barrier reflected BDSDEs.in such case, we confirmed the truth of that a saddle point of the stochastic recursive differential game problem above-mentioned is existed and obtained the optimal payoff functional.

I. Introduction

Pardoux and Peng [1] first introduced the nonlinear backward stochastic differential equations (BSDEs in short) and gave a proof about the adapted solutions is existed and unique under some hypothesis. Then Duffie and Epstein [2] studied the BSDE in the views of economic, they introduced a recursive utility, i.e. it corresponds to the solution of a particular BSDE and the BSDE's generator doesn't rely on the variable Z. the result in [1] is more general in the view of the mathematical point. Afterwards, respectively, El Karoui et al. [3] and Cvitanic and Karatzas [4] extended the results of BSDES to the backward stochastic differential equations with reflection, or it could be expressed with the concept of “barrier”.