Relationship between backward and forward linear-quadratic mean-field-game with terminal constraint and optimal asset allocation for insurers and pension funds

Herein, motivated by problems faced by insurance firms, we consider the dynamic games of N weakly coupled linear forward stochastic systems with terminal constraints involving mean-field interactions. By penalisation method, the associated mean-field game (MFG) is formulated and its consistency condition is given by a fully coupled forward–backward stochastic differential equation (FBSDE). Moreover, the decentralised strategies are obtained, and the ε-Nash equilibrium is verified. In addition, we study the connection of backward linear quadratic (LQ) MFG and forward LQ MFG with terminal constraint. Furthermore, the decoupled optimal strategies of this MFG are solved explicitly by introducing some Riccati equations. As an illustration, some simulations of the optimal asset allocation for the firm and pension funds are further studied.