On Numerical approximations of fractional and nonlocal Mean Field Games

TL;DR

This paper develops and analyzes numerical schemes for solving Mean Field Games with fractional or nonlocal diffusions, demonstrating their stability, convergence, and effectiveness through theoretical proofs and numerical tests.

Contribution

It introduces monotone, stable, and consistent semi-Lagrangian schemes for nonlocal Mean Field Games, with proven convergence and applicability to various diffusion types.

Findings

Schemes are monotone, stable, and consistent.

Convergence is proven for degenerate and nondegenerate cases.

Numerical tests confirm theoretical results.

Abstract

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.