Viscosity-variational inequality solutions of fully nonlinear second-order stationary mean field games with nonseparable and nondifferentiable Hamiltonians

Abstract

We analyse stationary fully nonlinear uniformly elliptic second-order mean field games with nonseparable and nondifferentiable Hamiltonians by introducing the notion of viscosity-variational inequality solution. In this formulation of the problem, the value function is the viscosity solution of the Hamilton--Jacobi--Bellman (HJB) equation, and the Kolmogorov--Fokker--Planck (KFP) equation for the player density is understood as a nonstandard variational inequality that no longer involves partial derivatives of the value function. The advantages of this formulation are found in its stability and continuity properties for handling weak convergence arguments, which we use to prove the existence of solutions of the problem. Furthermore, we prove the equivalence of the notion of viscosity-variational inequality solution with the notion of a partial differential inclusion solution, where the value function is a strong solution of the HJB equation and the player density is a very weak solution of the KFP equation with coefficients belonging to subdifferential sets of the Hamiltonian. Finally, we prove the uniqueness of solutions in the case of separable Hamiltonians with strictly monotone couplings.