| Abstract: |
| The Mean Field Game (MFG) system of Partial Differential Equations (PDE), introduced by Lasry \& Lions in 2006, models Nash equilibria of large population stochastic differential games of optimal control where the players of the game have unique optimal controls, and the convex Hamiltonian of the underlying optimal control problem is differentiable. In this talk, we introduce a new class of model problems called Mean Field Game Partial Differential Inclusions (MFG PDI), which extend the MFG system of Lasry and Lions to situations where players may have possibly nonunique optimal controls, and the resulting Hamiltonian of the underlying optimal control problem is not required to be differentiable. We prove the existence of unique weak solutions to MFG PDI for a broad class of Hamiltonians that are convex, Lipschitz, but possibly nondifferentiable, under a monotonicity condition similar to one considered previously by Lasry \& Lions. Moreover, we introduce a class of monotone finite element discretizations of the weak formulation of MFG PDI and present theorems on the strong convergence of the approximations to the value function in the $L^2(H_0^1)$-norm and the strong convergence of the approximations to the density function in $L^p(L^2)$-norms. We conclude the talk with discussion of numerical experiments involving non-smooth solutions. |
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