A mean-field version of Bank-El Karoui's representation of stochastic processes

TL;DR

This paper extends Bank-El Karoui's stochastic process representation to a mean-field setting, establishing existence, uniqueness, and stability results, with applications to various Mean-Field Games involving common noise and multiple populations.

Contribution

It introduces a mean-field version of Bank-El Karoui's theorem, providing a unified framework for analyzing complex Mean-Field Games with new stability insights.

Findings

Established existence and uniqueness under various conditions

Provided a stability theorem for the classical representation

Applied results to multiple Mean-Field Game models

Abstract

We study a mean-field version of Bank-El Karoui's representation theorem of stochastic processes. Under different technical conditions, we establish some existence and uniqueness results. As motivation and first applications, our mean-field representation results provide a unified approach to study different Mean-Field Games (MFGs) in the setting with common noise and multiple populations, including the MFG of timing, the MFG with singular control, etc. As a crucial technical step, we provide a stability result on the classical Bank-El Karoui's representation theorem, which has its own interests and other applications, such as in deriving stability results of the optimizers (in the strong sense) for a class of optimal stopping problems and singular control problems.