  This paper examines mean field linear-quadratic-Gaussian (LQG) social optimum
control with volatility-uncertain common noise. The diffusion terms in the
dynamics of agents contain an unknown volatility process driven by a common
noise. We apply a robust optimization approach in which all agents view
volatility uncertainty as an adversarial player. Based on the principle of
person-by-person optimality and a two-step-duality technique for stochastic
variational analysis, we construct an auxiliary optimal control problem for a
representative agent. Through solving this problem combined with a consistent
mean field approximation, we design a set of decentralized strategies, which
are further shown to be asymptotically social optimal by perturbation analysis.
