A Semi-Lagrangian scheme for a degenerate second order Mean Field Game system

TL;DR

This paper develops and analyzes a Semi-Lagrangian numerical scheme for a second order, possibly degenerate, Mean Field Game system, demonstrating convergence in one-dimensional cases and providing numerical evidence of effectiveness.

Contribution

It introduces a fully discrete Semi-Lagrangian scheme for degenerate second order Mean Field Games and proves its well-posedness and convergence in one dimension.

Findings

Scheme is well posed

Convergence proven in 1D cases

Numerical simulations confirm effectiveness

Abstract

In this paper we study a fully discrete Semi-Lagrangian approximation of a second order Mean Field Game system, which can be degenerate. We prove that the resulting scheme is well posed and, if the state dimension is equals to one, we prove a convergence result. Some numerical simulations are provided, evidencing the convergence of the approximation and also the difference between the numerical results for the degenerate and non-degenerate cases.