Second Order Fully Nonlinear Mean Field Games with Degenerate Diffusions
Alain Bensoussan, Ziyu Huang, Shanjian Tang, Sheung Chi Phillip Yam
TL;DR
The paper tackles the global-in-time well-posedness of second-order mean field games with nonlinear drifts and degenerate, distribution-dependent diffusion, not depending on the control. It develops a probabilistic framework based on forward–backward SDEs driven by a β-monotonicity structure, enabling existence, uniqueness, and regularity results under relatively weak assumptions. By analyzing Jacobian and Hessian flows, the authors establish Gâteaux and Hessian differentiability of the state–adjoint system with respect to initial data and measure, and show the value function V is a classical solution to the mean field master equation, with explicit characterizations of its spatial and distributional derivatives. The approach directly imposes conditions on the drift, diffusion, and costs, accommodates degenerate and nonlinearly distribution-dependent diffusion, and yields a robust pathway to master-equation solvability, with potential extensions to common-noise and mean-field type control problems.
Abstract
In this article, we study the global-in-time well-posedness of second order mean field games (MFGs) with both nonlinear drift functions simultaneously depending on the state, distribution and control variables, and the diffusion term depending on both state and distribution. Besides, the diffusion term is allowed to be degenerate, unbounded and even nonlinear in the distribution, but it does not depend on the control. First, we establish the global well-posedness of the corresponding forward-backward stochastic differential equations (FBSDEs), which arise from the maximum principle under a so-called $β$-monotonicity commonly used in the optimal control theory. The $β$-monotonicity admits more interesting cases, as representative examples including but not limited to the displacement monotonicity, the small mean field effect condition or the Lasry-Lions monotonicity; and ensures the well-posedness result in diverse non-convex examples. In our settings, we pose assumptions directly on the drift and diffusion coefficients and the cost functionals, rather than indirectly on the Hamiltonian, to make the conditions more visible. Our probabilistic method tackles the nonlinear dynamics with a linear but infinite dimensional version, and together with our recently proposed cone property for the adjoint processes, following in an almost straightforward way the conventional approach to the classical stochastic control problem, we derive a sufficiently good regularity of the value functional, and finally show that it is the unique classical solution to the MFG master equation. Our results require fairly few conditions on the functional coefficients for solution of the MFG, and a bit more conditions -- which are least stringent in the contemporary literature -- for classical solution of the MFG master equation.