A semi-Lagrangian method for solving state constraint Mean Field Games in Macroeconomics
Fabio Camilli, Qing Tang, Yong-shen Zhou
TL;DR
This work develops a semi-Lagrangian framework for solving state-constrained mean field games arising in macroeconomic models of heterogeneous agents (Aiyagari–Bewley–Huggett). It couples a constrained Hamilton–Jacobi–Bellman equation with a Fokker–Planck–Kolmogorov equation and uses a dual SL scheme for the FP, along with Howard's policy iteration to handle borrowing constraints and an endogenously determined interest rate $r$. A strong comparison principle for constrained viscosity solutions is established, and convergence of the SL scheme to the constrained solution is proved via Barles–Souganidis arguments; the authors also provide fully discrete, implementable algorithms and demonstrate their approach on stationary and transition scenarios, capturing wealth distributions and policy responses. The results yield a robust computational tool for equilibria in continuous-time MFG macro models with state constraints, enabling reliable simulations of wealth dynamics and macroeconomic policy effects.
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.