Regularity of the Value Function in Discounted Infinite-Time Mean Field Games

TL;DR

This paper investigates the regularity properties of the value function in discounted infinite-time mean field games, establishing existence, uniqueness, and differentiability results using FBSDE techniques.

Contribution

It proves strong existence and uniqueness for a broad class of infinite-time FBSDEs and characterizes the measure derivative of the value function via FBSDE solutions.

Findings

Proved strong existence and uniqueness of solutions for extended infinite-time FBSDEs

Established Lions-differentiability of the value function with respect to the measure

Provided explicit characterization of the derivative using FBSDE solutions

Abstract

In [17], we introduced the discounted infinite-time mean field games. Subsequently, in [18], we studied the connection between infinite-time mean field FBSDEs and elliptic master equations. In this paper, we further investigate the regularity of the representative player's value function. Specifically, we first prove the strong existence and uniqueness, as well as the uniqueness in law, for an extended class of infinite-time FBSDEs. We then establish the Lions-differentiability for the derivative of the representative player's value function with respect to the measure argument, and provide an explicit characterization for it using solutions to FBSDEs.