Social Optima in Mean Field Linear-Quadratic-Gaussian Control with Volatility Uncertainty

TL;DR

This paper develops decentralized control strategies for large populations of agents under volatility uncertainty in a mean field LQG setting, ensuring asymptotic social optimality through robust optimization and duality techniques.

Contribution

It introduces a robust optimization framework for mean field LQG control with unknown volatility, and constructs decentralized strategies that are asymptotically social optimal.

Findings

Decentralized strategies are asymptotically social optimal.

Robust control approach handles volatility uncertainty effectively.

Duality techniques facilitate the solution of the control problem.

Abstract

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.