Direct Approach of Indefinite Linear-Quadratic Mean Field Games

This paper is concerned with an indefinite linear-quadratic (LQ) mean field games of stochastic large-polulation system, where the individual diffusion coefficients can depend on both the state and the control of the agents and the population state average. Moreover, the control weights in the cost functionals could be indefinite. We use a direct approach to derive the -Nash equilibrium strategy. First, we formally solving an N-player game problem within a vast and finite population setting. Subsequently, decoupling or reducing high-dimensional systems by introducing two Riccati equations explicitly yields centralized strategies, contingent on the state of a specific player and the average state of the population. As the population size N approaches infinity, the construction of decentralized strategies becomes feasible. Then, we demonstrated they are an -Nash equilibrium.