Mean field games with partially observed major player and stochastic mean field

Mean field game (MFG) theory where there is a major player and many minor players (MM-MFG) has been recently introduced in both the linear quadratic Gaussian (LQG) case and in the case of nonlinear state dynamics and nonlinear cost functions. In this framework, a major player has a significant influence, i.e., asymptotically non-vanishing, on any minor agent. In contrast to the situation without major agents, the mean field term now becomes stochastic due to the stochastic evolution of the state of the major player and, as a result, the best response control actions of the minor agents depend on the state of the major agent as well as the stochastic mean field. In a decentralized environment, one is led to consider the situation where the agents are provided only with partial information on the major agent´s state and the mean field term. In this work, we consider such a scenario for systems with nonlinear dynamics and cost functions and develop MFG theory for a partially observed MM-MFG setup. More explicitly, we consider a MFG problem with (i) partial observations on the major player state provided to the minor agents individually and (ii) complete observations on that state provided to the major player. The first step of such a theory requires one to develop an estimation theory for partially observed stochastic dynamical systems whose state equations are of McKean-Vlasov (MV) type stochastic differential equations. The next approach to the problem for MM-MFG systems in this work is to follow the procedure of constructing the associated completely observed system via the application of nonlinear filtering theory. The existence and uniqueness of Nash equilibria is then analyzed in this setting.