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The global well-posedness for master equations of mean field games of controls

Shuhui Liu, Xintian Liu, Chenchen Mou, Defeng Sun

TL;DR

The paper addresses global well-posedness for master equations of mean field games of controls with joint state-control law interaction. It develops integral forms of Lasry-Lions and displacement λ-monotonicity, proves their propagation through the MFGC system, and derives a priori Lipschitz estimates in the measure variable. These steps yield a global classical solution to the master equation under either monotonicity regime, including nonseparable Hamiltonians and reduced regularity. The results extend the theory of MFG master equations by providing general global well-posedness for MFGC master equations and establishing key Lipschitz properties that underpin stability and convergence analyses.

Abstract

In this manuscript, we establish the global well-posedness for master equations of mean field games of controls, where the interaction is through the joint law of the state and control. Our results are proved under two different conditions: the Lasry-Lions monotonicity and the displacement $λ$-monotonicity, both considered in their integral forms. We provide a detailed analysis of both the differential and integral versions of these monotonicity conditions for the corresponding nonseparable Hamiltonian and examine their relation. The proof of global well-posedness relies on the propagation of these monotonicity conditions in their integral forms and a priori uniform Lipschitz continuity of the solution with respect to the measure variable.

The global well-posedness for master equations of mean field games of controls

TL;DR

The paper addresses global well-posedness for master equations of mean field games of controls with joint state-control law interaction. It develops integral forms of Lasry-Lions and displacement λ-monotonicity, proves their propagation through the MFGC system, and derives a priori Lipschitz estimates in the measure variable. These steps yield a global classical solution to the master equation under either monotonicity regime, including nonseparable Hamiltonians and reduced regularity. The results extend the theory of MFG master equations by providing general global well-posedness for MFGC master equations and establishing key Lipschitz properties that underpin stability and convergence analyses.

Abstract

In this manuscript, we establish the global well-posedness for master equations of mean field games of controls, where the interaction is through the joint law of the state and control. Our results are proved under two different conditions: the Lasry-Lions monotonicity and the displacement -monotonicity, both considered in their integral forms. We provide a detailed analysis of both the differential and integral versions of these monotonicity conditions for the corresponding nonseparable Hamiltonian and examine their relation. The proof of global well-posedness relies on the propagation of these monotonicity conditions in their integral forms and a priori uniform Lipschitz continuity of the solution with respect to the measure variable.
Paper Structure (12 sections, 10 theorems, 90 equations)

This paper contains 12 sections, 10 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The product probability spaces and function spaces
  4. Mean field games of controls
  5. The master equation and the MFGC system
  6. Monotonicities and main assumptions
  7. Relation between forms of monotonicities for the Hamiltonian
  8. Propagation of the monotonicities
  9. The uniform Lipschitz continuity in the measure variable
  10. The W2-Lipschitz continuity under displacement lambda-monotonicity condition
  11. The W1-Lipschitz continuity under Lasry-Lions monotonicity condition
  12. Global well-posedness of the master equation

Key Result

Proposition 2.11

Let Assumptions G regularity, H regularity(i)(ii) hold and $\rho:[0,T]\times \Omega\to\mathcal{P}_2(\mathop{\mathrm{\mathbb{R}}}\nolimits^{2d})$ be $\mathbb{F}^0$-progressively measurable with (i) For any $x\in\mathbb{R}^d$ and for the $X^x$ in (decoupled FBSDE), the following BSDE on $[t_0,T]$ has a unique solution with bounded $Z^x$: where $\pi_1\#\rho=\mu$. (ii) Denote $u(t_0,x)=Y_{t_0}^x$. T

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.8
  • Remark 2.10
  • Proposition 2.11: mou2022displacement
  • Lemma 3.3
  • proof
  • ...and 19 more