Convergence of a Finite Difference Scheme to Weak Solutions of the System of Partial Differential Equations Arising in Mean Field Games

Mean field--type models describing the limiting behavior of stochastic differential games as the number of players tends to +∞ were recently introduced by Lasry and Lions. Under suitable assumptions, they lead to a system of two coupled partial differential equations, a forward Bellman equation and a backward Fokker--Planck equation. Finite difference schemes for the approximation of such systems have been proposed in previous works. Here, we prove the convergence of these schemes towards a weak solution of the system of partial differential equations.