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Mean field games with common noise via Malliavin calculus

Ludovic Tangpi, Shichun Wang

TL;DR

This work studies mean field games with common noise, where players interact through the conditional law $m_t=\mathcal{L}^{\mathbb{P}}(X_t|X^c_t)$. It provides a simpler existence proof of strong equilibria by extending Malliavin-compactness to process-valued mappings and employing a mimicking argument to realize a Markovian, strongly adapted equilibrium on the original probability space, without requiring monotonicity or high regularity. The framework only requires measurability in the state for drift and cost functionals, and yields a Markovian optimal control under a weak formulation that is then upgraded to a strong formulation. The results broaden the applicability of MFGs with common noise and offer a robust, discretization-free approach applicable to models such as financial optimal execution, while potentially yielding non-unique equilibria due to the relaxed assumptions.

Abstract

We present a simpler proof of the existence of equilibria for a class of mean field games with common noise, where players interact through the conditional law given the current value of the common noise rather than its entire path. By extending a compactness criterion for Malliavin-differentiable random variables to processes, we establish existence of strong equilibria, where the conditional law and optimal control are adapted to the common noise filtration and defined on the original probability space. Notably, our approach only requires measurability of the drift and cost functionals with respect to the state variable.

Mean field games with common noise via Malliavin calculus

TL;DR

This work studies mean field games with common noise, where players interact through the conditional law . It provides a simpler existence proof of strong equilibria by extending Malliavin-compactness to process-valued mappings and employing a mimicking argument to realize a Markovian, strongly adapted equilibrium on the original probability space, without requiring monotonicity or high regularity. The framework only requires measurability in the state for drift and cost functionals, and yields a Markovian optimal control under a weak formulation that is then upgraded to a strong formulation. The results broaden the applicability of MFGs with common noise and offer a robust, discretization-free approach applicable to models such as financial optimal execution, while potentially yielding non-unique equilibria due to the relaxed assumptions.

Abstract

We present a simpler proof of the existence of equilibria for a class of mean field games with common noise, where players interact through the conditional law given the current value of the common noise rather than its entire path. By extending a compactness criterion for Malliavin-differentiable random variables to processes, we establish existence of strong equilibria, where the conditional law and optimal control are adapted to the common noise filtration and defined on the original probability space. Notably, our approach only requires measurability of the drift and cost functionals with respect to the state variable.
Paper Structure (6 sections, 10 theorems, 65 equations)

This paper contains 6 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Setting and Main Result
  3. Existence of MFE: Proof of Theorem \ref{['thm:main']}
  4. Weak Formulation of MFG
  5. Malliavin Calculus
  6. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 2.5

Under Assumptions Assumption:E and Assumption:M, the mean field game admits a mean field equilibrium $(\hat{m}, \hat{\alpha}) \in \mathcal{M} \times \mathbb{A}$. Moreover, its optimal control is Markovian. Specifically, there exists a measurable function $\hat{\alpha}_M: [0, T] \times \mathbb{R}^{d_ where $(X^{\hat{\alpha}}, X^{c})$ uniquely solves the SDE

Theorems & Definitions (22)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.5
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Definition 3.4
  • Proposition 3.5
  • proof
  • Corollary 3.6
  • ...and 12 more