A Semi-Lagrangian Method for Solving State Constraint Mean Field Games in Macroeconomics

Abstract. We study continuous-time heterogeneous agent models cast as Mean Field Games, in the Aiyagari–Bewley–Huggett framework. The model couples a Hamilton–Jacobi–Bellman equation for individual optimization with a Fokker–Planck–Kolmogorov equation for the wealth distribution. We establish a comparison principle for constrained viscosity solutions of the HJB equation and propose a semi-Lagrangian (SL) scheme for its numerical solution, proving convergence via the Barles–Souganidis method. A policy iteration algorithm handles state constraints, and a dual SL scheme is used for the FPK equation. Numerical methods are presented in a fully discrete, implementable form.