\epsilon-Nash Mean Field Game Theory for Nonlinear Stochastic Dynamical Systems with Major and Minor Agents

This paper studies a large population dynamic game involving nonlinear stochastic dynamical systems with agents of the following mixed types: (i) a major agent, and (ii) a population of $N$ minor agents where $N$ is very large. The major and minor (MM) agents are coupled via both: (i) their individual nonlinear stochastic dynamics, and (ii) their individual finite time horizon nonlinear cost functions. This problem is approached by the so-called $\epsilon$-Nash Mean Field Game ($\epsilon$-NMFG) theory. A distinct feature of the mixed agent MFG problem is that even asymptotically (as the population size $N$ approaches infinity) the noise process of the major agent causes random fluctuation of the mean field behaviour of the minor agents. To deal with this, the overall asymptotic ($N \rightarrow \infty$) mean field game problem is decomposed into: (i) two non-standard stochastic optimal control problems with random coefficient processes which yield forward adapted stochastic best response control processes determined from the solution of (backward in time) stochastic Hamilton-Jacobi-Bellman (SHJB) equations, and (ii) two stochastic coefficient McKean-Vlasov (SMV) equations which characterize the state of the major agent and the measure determining the mean field behaviour of the minor agents. Existence and uniqueness of the solution to the Stochastic Mean Field Game (SMFG) system (SHJB and SMV equations) is established by a fixed point argument in the Wasserstein space of random probability measures. In the case that minor agents are coupled to the major agent only through their cost functions, the $\epsilon_N$-Nash equilibrium property of the SMFG best responses is shown for a finite $N$ population system where $\epsilon_N=O(1/\sqrt N)$.