Finite Horizon Mean Field Games on Networks

TL;DR

This paper studies finite horizon stochastic mean field games on networks, establishing existence and uniqueness of solutions with complex boundary conditions for the value function and density.

Contribution

It introduces a framework for mean field games on networks with Kirchhoff and transmission conditions, proving well-posedness for Lipschitz Hamiltonians.

Findings

Existence of solutions for the coupled HJB-Fokker-Planck system.

Uniqueness of solutions under Lipschitz continuity.

Characterization of boundary conditions at network vertices.

Abstract

We consider finite horizon stochastic mean field games in which the state space is a network. They are described by a system coupling a backward in time Hamilton-Jacobi-Bellman equation and a forward in time Fokker-Planck equation. The value function u is continuous and satisfies general Kirchhoff conditions at the vertices. The density m of the distribution of states satisfies dual transmission conditions: in particular, m is generally discontinuous across the vertices, and the values of m on each side of the vertices satisfy special compatibility conditions. The stress is put on the case when the Hamiltonian is Lipschitz continuous. Existence and uniqueness are proven.