Differential games of partial information forward-backward doubly stochastic differential equations and applications

This paper is concerned with a new type of differential game problems of forward-backward stochastic systems. There are three distinguishing features: Firstly, our game systems are forward-backward doubly stochastic differential equations, which is a class of more general game systems than other forward-backward stochastic game systems without doubly stochastic terms; Secondly, forward equations are directly related to backward equations at initial time, not terminal time; Thirdly, the admissible control is required to be adapted to a sub-information of the full information generated by the underlying Brownian motions. We give a necessary and a sufficient conditions for both an equilibrium point of nonzero-sum games and a saddle point of zero-sum games. Finally, we work out an example of linear-quadratic nonzero-sum differential games to illustrate the theoretical applications. Applying some stochastic filtering techniques, we obtain the explicit expression of the equilibrium point.