A kind of non-zero sum mixed differential game of backward stochastic differential equation

This paper is concerned with a non-zero sum mixed differential game problem described by a backward stochastic differential equation. Here the term "mixed" means that this game problem contains a deterministic control v1 of Player 1 and a random control process v2 of Player 2. By virtue of the classical variational method, a necessary condition and an Arrow's sufficient condition for the mixed stochastic differential game problem are presented. A linear-quadratic mixed differential game problem is discussed, and the corresponding Nash equilibrium point is explicitly expressed by the solution of mean-field forward-backward stochastic differential equation. The most distinguishing feature, compared with the existing literature, is that the optimal state process of the linear-quadratic game satisfies a linear mean-field backward stochastic differential equation. Finally, a home mortgage and wealth management problem is given to illustrate our theoretical results.