A Control Theoretical Approach to Mean Field Games and Associated Master Equations

Abstract: We prove the global-in-time well-posedness for a broad class of mean field game problems, which is beyond the special linear-quadratic setting, as long as the mean field sensitivity is not too large. Through the stochastic maximum principle, we adopt the FBSDE approach to investigate the unique existence of the corresponding equilibrium strategies. The corresponding FBSDEs are first solved locally in time, then by controlling the sensitivity of the backward solutions with respect to the initial condition via some suitable apriori estimates for the corresponding Jacobian flows, the global-in-time solution is warranted. Further analysis on these Jacobian flows will be discussed to establish the regularities, such as linear functional differentiability, of the respective value functions that leads to the ultimate classical well-posedness of the master equation on $\mathbb{R}^d$. To the best of our knowledge, it is the first article to deal with the mean field game problem, as well as its associated master equation, with general cost functionals having quadratic growth under the small mean field effect. In this current approach, we directly impose the structural conditions on the cost functionals, rather than conditions on the Hamiltonian. The advantages of this are threefold: (i) compared with imposing conditions on Hamiltonian, the structural conditions imposed in this work are easily verified, and less demanding on the regularity requirements of the cost functionals while solving the master equation; (ii) the displacement monotonicity is basically just a direct consequence of small mean field effect in the structural conditions; and (iii) when the mean field effect is not that small, we can still provide an accurate lifespan for the local existence. The method in this work can be readily extended to the case with nonlinear drift and non-separable cost functionals.