Forward-backward stochastic differential equations with monotone functionals and mean field games with common noise

TL;DR

This paper establishes existence, uniqueness, and estimates for forward-backward stochastic differential equations with monotone functionals, and applies these results to mean field games with common noise, providing conditions for well-posedness and feedback control solutions.

Contribution

It introduces new existence and uniqueness results for FBSDEs with monotone functionals and applies them to solve mean field games with common noise, including feedback control characterization.

Findings

Existence and uniqueness of FBSDEs with monotone functionals.

Well-posedness of mean field FBSDEs with conditional law.

Existence of feedback form optimal controls in mean field games.

Abstract

In this paper, we consider a system of forward-backward stochastic differential equations (FBSDEs) with monotone functionals. We show the existence and uniqueness of such a system by the method of continuation similarly to Peng and Wu (1999) for classical FBSDEs and obtain estimates under conditional probability. As applications, we prove the well-posedness result for a mean field FBSDE with conditional law and show the existence of a decoupling function. In addition, we show that mean field games with common noise are uniquely solvable under a linear controlled process with convex and weak-monotone cost functions and prove that the optimal control is in a feedback form depending only on the current state and conditional law.