Abstract
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 v_{1} of Player 1 and a random control process v_{2} 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.
Highlights
Differential game theory involves multiple individuals decision making in the context of dynamical systems
The study of differential game was originally stated by Isaacs [1], and summed up and developed by Basar and Olsder [2]
Hamadène and Lepeltier [5] discussed a stochastic zero sum differential game problem, and investigated the existence of saddle point under Isaacs’ condition. Their results of this problem depend on the solution of backward stochastic differential equation (BSDE)
Summary
Differential game theory involves multiple individuals ( called players or agents) decision making in the context of dynamical systems. Hamadène and Lepeltier [5] discussed a stochastic zero sum differential game problem, and investigated the existence of saddle point under Isaacs’ condition. Their results of this problem depend on the solution of BSDE. Zhang Advances in Difference Equations (2020) 2020:37 studied a new non-zero sum stochastic differential game of BSDE, and established a necessary condition and a sufficient condition in the form of a maximum principle for an open-loop equilibrium point. Wang et al [9] discussed asymmetric information linear–quadratic (LQ) non-zero sum differential game of BSDE, and gave the feedback Nash equilibrium points.
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