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Second-order monotonicity conditions and mean field games with volatility control

Chenchen Mou, Jianfeng Zhang, Jianjun Zhou

TL;DR

This work develops a theory for the master equation of mean field games with volatility control, where the equation is fully nonlinear in both $\partial_x V$ and $\partial_{xx}V$. It introduces second-order Lasry-Lions monotonicity, proves its propagation under a separable Hamiltonian, and establishes local and global well-posedness for the master equation via a rigorous McKean-Vlasov FBSDE framework. The paper shows that propagation of monotonicity yields uniform $W_1$-Lipschitz bounds in the measure argument, enabling global well-posedness by gluing local solutions. It also provides stability and a detailed local regularity theory, including a representation formula for $\partial_\mu V$, which is pivotal for understanding how the value function depends on the distribution of states. These results extend master equation theory to MFGs with volatility control and lay groundwork for further extensions to non-separable Hamiltonians and second-order displacement monotonicity.

Abstract

In this manuscript we study the well-posedness of the master equations for mean field games with volatility control. This infinite dimensional PDE is nonlinear with respect to both the first and second-order derivatives of its solution. For standard mean field games with only drift control, it is well-known that certain monotonicity condition is essential for the uniqueness of mean field equilibria and for the global well-posedness of the master equations. To adapt to the current setting with volatility control, we propose a new notion called second-order monotonicity conditions. Surprisingly, the second-order Lasry-Lions monotonicity is equivalent to its standard (first-order) version, but such an equivalency fails for displacement monotonicity. When the Hamiltonian is separable and the data are Lasry-Lions monotone, we show that the Lasry-Lions monotonicity propagates and the master equation admits a unique classical solution. This is the first work for the well-posedness, both local and global, of master equations when the volatility is controlled.

Second-order monotonicity conditions and mean field games with volatility control

TL;DR

This work develops a theory for the master equation of mean field games with volatility control, where the equation is fully nonlinear in both and . It introduces second-order Lasry-Lions monotonicity, proves its propagation under a separable Hamiltonian, and establishes local and global well-posedness for the master equation via a rigorous McKean-Vlasov FBSDE framework. The paper shows that propagation of monotonicity yields uniform -Lipschitz bounds in the measure argument, enabling global well-posedness by gluing local solutions. It also provides stability and a detailed local regularity theory, including a representation formula for , which is pivotal for understanding how the value function depends on the distribution of states. These results extend master equation theory to MFGs with volatility control and lay groundwork for further extensions to non-separable Hamiltonians and second-order displacement monotonicity.

Abstract

In this manuscript we study the well-posedness of the master equations for mean field games with volatility control. This infinite dimensional PDE is nonlinear with respect to both the first and second-order derivatives of its solution. For standard mean field games with only drift control, it is well-known that certain monotonicity condition is essential for the uniqueness of mean field equilibria and for the global well-posedness of the master equations. To adapt to the current setting with volatility control, we propose a new notion called second-order monotonicity conditions. Surprisingly, the second-order Lasry-Lions monotonicity is equivalent to its standard (first-order) version, but such an equivalency fails for displacement monotonicity. When the Hamiltonian is separable and the data are Lasry-Lions monotone, we show that the Lasry-Lions monotonicity propagates and the master equation admits a unique classical solution. This is the first work for the well-posedness, both local and global, of master equations when the volatility is controlled.

Paper Structure

This paper contains 19 sections, 10 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Second-order monotonicity conditions
  3. The basic setting
  4. Second-order Lasry-Lions monotonicity condition
  5. Second-order displacement monotonicity condition
  6. Mean field games with volatility control
  7. The main results
  8. The stability and the local well-posedness
  9. The global well-posedness
  10. The global well-posedness: Proof of Theorem \ref{['thm-global']}
  11. Propagation of monotonicity
  12. A priori $W_1$-Lipschitz continuity
  13. Proof of Theorem \ref{['thm-global']}.
  14. The stability: Proof of Theorem \ref{['thm-stability']}
  15. The local well-posedness: Proof of Theorem \ref{['thm-local']}.
  16. ...and 4 more sections

Key Result

Proposition 2.3

Let $U\in \mathcal{C}_b^{1,4}(\mathbb R\times\mathcal{P}_2(\mathbb{R}))$. Then $U$ is Lasry-Lions monotone if and only if it satisfies the second-order Lasry-Lions monotonicity condition.

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 3.1
  • Remark 4.3
  • Remark 4.4
  • Theorem 4.5
  • ...and 10 more