On Time-Inconsistent Extended Mean-Field Control Problems with Common Noise

This paper studies a class of time-inconsistent mean field control (MFC) problems in the presence of common noise under nonexponential 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 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.

Funding: Z. Liang is supported by the National Natural Science Foundation of China [Grant 12271290]. X. Yu and K. Zhang are supported by the Hong Kong Research Grants Council General Research Fund [Grants 15306523 and 15211524] and by the Research Centre for Quantitative Finance at the Hong Kong Polytechnic University [Grant P0042708].