  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
$\epsilon_N=O(1/\sqrt N)$.
