In this study, we consider a class of asymmetric information linear-quadratic non-zero sum stochastic differential game problems. The system dynamics are governed by a forward linear mean-field stochastic differential equation and the cost functional is quadratic. In order to facilitate some practical applications, the mean-field terms for the state process and control process are considered in both the system dynamics and cost functional. Using the classical calculus of variation and dual methods, the open-loop Nash equilibrium point can be expressed by introducing an auxiliary mean-field forward backward stochastic differential equation, which comprises one forward and two backward components. Due to some Riccati equations and ordinary differential equations that possess unique solutions, the Nash equilibrium point can also be represented in feedback form for several special cases under asymmetric information. The corresponding filtering equations are derived, and the existence and uniqueness of the solutions are proved. An investment problem in finance is discussed to demonstrate the good performance of the theoretical results.

在这项研究中，我们考虑了一类非对称信息线性二次非零和随机微分博弈问题。系统动力学由一个正向线性平均场随机微分方程控制，成本函数为二次方。为了方便一些实际应用，在系统动力学和成本函数中都考虑了状态过程和控制过程的平均场项。使用经典的变分演算和对偶方法，可以通过引入辅助平均场正向后向随机微分方程来表示开环Nash平衡点，该方程包括一个前向分量和两个后向分量。由于某些Riccati方程和具有独特解的常微分方程，对于不对称信息下的几种特殊情况，纳什均衡点也可以以反馈形式表示。推导了相应的滤波方程，证明了解的存在性和唯一性。讨论了金融投资问题，以证明理论结果的良好性能。