On time-inconsistent extended mean-field control problems with common noise

TL;DR

This paper develops a framework for analyzing time-inconsistent mean-field control problems with common noise, characterizing equilibrium strategies via advanced PDEs and applying results to financial models.

Contribution

It introduces a novel approach to characterize time-consistent equilibria in extended mean-field control problems with common noise, including LQ and non-LQ cases.

Findings

Existence of time-consistent equilibria in LQ mean-field control problems.

Characterization of equilibrium strategies via Hamilton-Jacobi-Bellman equations on Wasserstein space.

Application to financial models demonstrating practical relevance.

Abstract

This paper studies a class of time-inconsistent mean field control (MFC) problems in the presence of common noise under non-exponential discount and joint law dependence of both state and control. We investigate the closed-loop time-consistent equilibrium strategies for these extended MFC problems and characterize them through an equilibrium Hamilton-Jacobi-Bellman (HJB) equation defined on the Wasserstein space. We first apply the results to the linear-quadratic (LQ) time-inconsistent MFC problems and obtain the existence of time-consistent equilibria via a comprehensive study of a nonlocal Riccati system. To illustrate the theoretical findings, two financial applications are presented. We then examine a class of non-LQ time-inconsistent MFC problems, for which we contribute the existence of time-consistent equilibria by analyzing a nonlocal nonlinear partial differential equation.