Linear-quadratic non-zero sum differential game for mean-field stochastic systems with asymmetric information

Abstract This paper is concerned with a class of asymmetric information linear-quadratic (LQ) non-zero sum stochastic differential game problems. The dynamic of system is governed by a forward linear mean-field stochastic differential equation (MF-SDE), and the cost functional is quadratic. Motivated by some practical applications, the mean-field term of state process and control process are taken into account in both system dynamic 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 (MF-FBSDE), which consists of one forward and two backward components. Owing to some Riccati equations and ordinary differential equations (ODEs), which possess the unique solutions, the Nash equilibrium point can also be represented in the feedback form for several special cases under asymmetric information. The corresponding filtering equations are derived and the existence and uniqueness of solutions are proved. An investment problem in finance is discussed to demonstrate the good performance of theoretical results.