Mean field games equations with quadratic Hamiltonian: a specific approach

TL;DR

This paper introduces a transformation and monotonic scheme for solving mean field games equations with quadratic Hamiltonians, enabling effective numerical solutions and experiments.

Contribution

It presents a novel change of variables and a monotonic scheme specifically for quadratic Hamiltonian mean field games, simplifying their analysis and computation.

Findings

Transformation simplifies MFG equations with quadratic Hamiltonian.

Monotonic scheme effectively constructs solutions.

Numerical experiments validate the proposed methods.

Abstract

Mean field games models describing the limit of a large class of stochastic differential games, as the number of players goes to $+\infty$, have been introduced by J.-M. Lasry and P.-L. Lions. We use a change of variables to transform the mean field games (MFG) equations into a system of simpler coupled partial differential equations, in the case of a quadratic Hamiltonian. This system is then used to exhibit a monotonic scheme to build solutions of the MFG equations. Effective numerical methods based on this constructive scheme are presented and numerical experiments are carried out.