Mean-Field Linear-Quadratic Nonzero Sum Stochastic Differential Games with Overlapping Information

Guided by stochastic control theory and filtering theory, this article employs classic stochastic differential equation theory, new decouple technology, and Pontryagin’s maximum principle with partial information to investigate a class of linear-quadratic (LQ) nonzero-sum differential game problems for mean-field stochastic systems with overlapping information. Here, “overlapping" refers to the existence of common elements in the information sets of the two players, without any inclusion relationship between them. The state equation is a forward stochastic differential equation, and a key difference from previous literature is that the diffusion term of the state equation contains the control variables of both players. Additionally, both the state equation and the cost functional take into account the mean-field terms of the state process and control process. Furthermore, the cost functional is quadratic. Firstly, by utilizing the maximum principle for partial information and introducing a backward stochastic differential equation, we obtain the necessary and sufficient conditions for the open-loop Nash equilibrium point. Subsequently, leveraging the existence and uniqueness of solutions to a system of coupled Riccati equations, we derive the feedback expression for the Nash equilibrium point, which provides a closed-loop representation of the open-loop solution. Moreover, we derive the corresponding filtering equations, which demonstrate the existence and uniqueness of the solution.