Linear quadratic mean-field game with volatility uncertainty

Abstract

In this paper, we consider a linear quadratic (LQ) mean field game (MFG) with uncertain volatility. Applying a robust optimization approach and based on mean-field approximation, we firstly obtain the worst uncertainty by dealing with an LQ MFG with so-called soft-constraint. Then with the help of convex analysis theory, we derive the auxiliary optimal control problems for the agents and construct the decentralized strategies. Finally, by virtue of estimates of (forward) backward stochastic differential equations, the asymptotical optimality of the decentralized strategy is verified. A numerical example is given to illustrate the above theoretical results.