Optimal Strategy of Mean-Field FBSDE Games with Delay and Noisy Memory Based on Malliavin Calculus

In this paper, we study a nonzero-sum stochastic differential game for a system whose dynamics are governed by a mean-field forward–backward stochastic differential equation (MF-FBSDE) with delay and noisy memory. We transform the stochastic game problem into a stochastic optimal control problem. By applying the variational method and using Malliavin calculus, we establish the sufficient and necessary maximum principles of controls, ensuring that all players are in Nash equilibrium in the game. Based on this, the optimal strategy for solving this kind of stochastic differential game problem is provided. We apply the result to finance, considering an optimal consumption model in a financial market that maximizes the recursive utility of each player in the Nash equilibrium. The numerical results illustrate the influences of mean-field and memory on the optimal recursive utilities of the players in the game. Furthermore, we provide a proof of the existence and uniqueness of the solution of this novel FBSDE system, and discuss the solution of its adjoint FBSDE system involving Malliavin derivatives.