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

Zongxia Liang, Xiang Yu, Keyu Zhang

TL;DR

This work develops a rigorous framework for time-inconsistent extended mean-field control with common noise, formulating a stochastic control problem where the conditional law given the common noise serves as the state. It derives an equilibrium Hamilton-Jacobi-Bellman equation on the Wasserstein space and proves a verification theorem linking equilibria to a value function. In the time-inconsistent LQ setting, the equilibrium reduces to a nonlocal Riccati system, and the authors establish well-posedness under standard positivity and monotonicity conditions. For general non-LQ problems, they employ a fixed-point strategy to analyze a nonlocal nonlinear PDE, proving global existence under a small-coupling assumption and connecting the PDE solution to closed-loop equilibria via the verification framework. Collectively, the results offer a comprehensive method to characterize and verify time-consistent equilibria in mean-field models with joint law dependence and common noise, with financial applications illustrating practical implications.

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.

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

TL;DR

This work develops a rigorous framework for time-inconsistent extended mean-field control with common noise, formulating a stochastic control problem where the conditional law given the common noise serves as the state. It derives an equilibrium Hamilton-Jacobi-Bellman equation on the Wasserstein space and proves a verification theorem linking equilibria to a value function. In the time-inconsistent LQ setting, the equilibrium reduces to a nonlocal Riccati system, and the authors establish well-posedness under standard positivity and monotonicity conditions. For general non-LQ problems, they employ a fixed-point strategy to analyze a nonlocal nonlinear PDE, proving global existence under a small-coupling assumption and connecting the PDE solution to closed-loop equilibria via the verification framework. Collectively, the results offer a comprehensive method to characterize and verify time-consistent equilibria in mean-field models with joint law dependence and common noise, with financial applications illustrating practical implications.

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.
Paper Structure (16 sections, 12 theorems, 215 equations)

This paper contains 16 sections, 12 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notations and preliminaries
  4. Sets and Functions
  5. Integration and probability
  6. Differentiability on Wassersetin space
  7. Time inconsistent extended MFC problems
  8. Characterization of Closed-loop Equilibrium Strategies
  9. Time-inconsistent LQ Extended MFC Problems
  10. A Class of Time-inconsistent Non-LQ Extended MFC Problems
  11. Technical Proofs
  12. The proof of Proposition \ref{['Pro:riccati']}
  13. The proof of Lemma \ref{['lemma:wellposedness:ricattiexample']}
  14. The proof of Proposition \ref{['newsect:lemma']}
  15. Step 1: Construction of the fixed point mapping
  16. ...and 1 more sections

Key Result

Proposition 2.1

Given a measurable space $(E,{\cal E})$ and a map $\rho$$:$$E$$\rightarrow$${\cal P}_{_2}(\mathbb{R}^d)$, $\rho$ is measurable if and only if the map $\langle\varphi,\rho\rangle$$:$$E$$\rightarrow$$\mathbb{R}$ is measurable, for any $\varphi\in\mathscr C_{_2}(\mathbb{R}^d;\mathbb{R})$.

Theorems & Definitions (29)

  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3: Closed-loop equilibrium strategy
  • Remark 2.4
  • Definition 2.5: Open-loop equilibrium strategy
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 4.1
  • ...and 19 more