ε-Nash Mean Field Game theory for nonlinear stochastic dynamical systems with mixed agents

We consider a large population dynamic game involving nonlinear stochastic dynamical systems with agents of the following mixed types: (i) a major agent, and (ii) a large N population of minor agents. The major and minor agents are coupled via both: (i) their individual nonlinear stochastic dynamics, and (ii) their individual finite time horizon nonlinear cost functions. We approach this problem by the so-called ε-Nash Mean Field Games (ε-NMFG) theory. In this problem 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 → ∞) mean field game problem is decomposed into: (i) two non-standard stochastic optimal control problems with random coefficient processes, 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. (i) and (ii) are coupled by the (forward adapted) stochastic best response processes determined from the solution of (backward in time) stochastic Hamilton-Jacobi-Bellman (SHJB) equations for the nonstandard optimal control problems in (i) which involve the state of the major agent and the distribution measure corresponding to the mean field behaviour of the minor agents in (ii) where these in turn depend upon the best response control processes themselves. When the so-called stochastic Mean Field (SMF) system (SHJB and SMV equations) is soluble we say the resulting framework is one of ε-NMFG type.