Convergence of a finite difference scheme to weak solutions of the system of partial differential equation arising in mean field games

TL;DR

This paper proves that a finite difference scheme converges to a weak solution of a coupled PDE system modeling mean field games, which describe the limiting behavior of large stochastic differential games.

Contribution

It establishes the convergence of existing finite difference schemes to weak solutions of the mean field game PDE system, filling a gap in the theoretical understanding.

Findings

Finite difference schemes converge to weak solutions.

Theoretical validation of numerical methods for mean field games.

Supports the use of these schemes in practical applications.

Abstract

Mean field type models describing the limiting behavior of stochastic differential games as the number of players tends to +$\infty$, have been recently introduced by J-M. Lasry and P-L. Lions. Under suitable assumptions, they lead to a system of two coupled partial differential equations, a forward Bellman equation and a backward Fokker-Planck equations. 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.