Mean Field Games with Mean-Field-Dependent Volatility: A PDE Approach

In this paper, we develop a PDE approach to consider the optimal strategy of mean field controlled stochastic system. Firstly, we discuss mean field SDEs and associated Fokker-Plank eqautions. Secondly, we consider a fully-coupled system of forward-backward PDEs. The backward one is the Hamilton-Jacobi-Bellman equation while the forward one is the Fokker-Planck equation. Our main result is to show the existence of classical solutions of the forward-backward PDEs in the class $H^{1+frac{1}{4},2+frac{1}{2}}([0,T]timesmathbb{R}^n)$ by use of the Schauder fixed point theorem. Then, we use the solution to give the optimal strategy of the mean field stochastic control problem. Finally, we give an example to illustrate the role of our main result.