  This paper investigates a class of mixed stochastic linear-quadratic-Gaussian
(LQG) social optimization and Nash game in the context of a large scale system.
Two types of interactive agents are involved: a major agent and a large number
of weakly-coupled minor agents. All minor agents are cooperative to minimize
the social cost as the sum of their individual costs, whereas such social cost
are conflictive to that of major agent. Thus, major agent and all minor agents
are further competitive to reach some non-zero Nash equilibrium. The control
processes enter both diffusion and drift terms of all major and minors' states.
This extends the standard setup in which control only enters the drift terms,
and such extension brings more modeling difference and technical difficulties,
in particular, when dealing with the feedback decentralized strategy via
Riccati equation and mean-field consistency condition (CC) representation.
Applying the mean-field approximations and person-by-person optimality, we
obtain auxiliary control problems for major agent and minor agents,
respectively. The decentralized social strategy is derived by a class of new CC
system, which is mean-field forward-backward stochastic differential equations.
The well-posedness of the CC system is obtained by the discounting method. The
related asymptotic social optimality for minor agents, and Nash equilibrium for
major-minor agents are also verified.
