The connection between mean-field linear-quadratic-Gaussian games of forward and backward stochastic differential systems

This paper is concerned with the connection between dynamic games of N weakly-coupled linear forward and backward stochastic differential equation systems involving mean-field interactions. The backward mean-field linear-quadratic-Gaussian (MFLQG) game is discussed here and its decentralized strategies is derived through the consistency condition. Next, the approximate Nash equilibrium of derived backward MFLQG game strategies are also proved. In addition, we study the connection of backward MFLQG game to a sequence of forward MFLQG games with controlled initial conditions and penalized terminal deviations. Under mild conditions, these two MFLQG game solutions are shown to be equivalent.