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Quantitative convergence rates for extended mean field games with volatility control

Erhan Bayraktar, Hiroaki Horikawa

TL;DR

This work addresses the quantitative convergence of open-loop Nash equilibria for symmetric $N$-player stochastic differential games with interactions through both states and controls, where volatility terms are controlled. It develops a stochastic maximum principle (SMP) framework and extends forward-backward propagation of chaos to the extended mean field setting, proving the well-posedness of conditional McKean–Vlasov FBSDEs via continuation under a displacement monotonicity condition. Under these assumptions, the paper establishes a rigorous $O(N^{-1})$ convergence rate between the $N$-player Nash equilibria and the mean field equilibrium, both in state trajectories and controls, by constructing a conditional MKV-FBSDE coupling and employing the method of continuation. To illustrate applicability, a linear-quadratic example is analyzed, showing unique existence of the MFE and validating the convergence rate, while avoiding a full closed-loop characterization. The results provide finite-$N$ guarantees for extended mean field games with volatility control, with potential applications in finance and economic modeling where control of volatility and control distribution are essential.

Abstract

We investigate the convergence of symmetric stochastic differential games with interactions via control, where the volatility terms of both idiosyncratic and common noises are controlled. We apply the stochastic maximum principle, following the approach of Laurière and Tangpi, to reduce the convergence analysis to the study of forward-backward propagation of chaos. Under the standard monotonicity conditions, we derive quantitative convergence rates for open-loop Nash equilibria of $N$-player stochastic differential games toward the corresponding mean field equilibrium. As a prerequisite, we also establish the well-posedness of the conditional McKean--Vlasov forward-backward stochastic differential equations by the method of continuation. Moreover, we analyze a specific class of linear-quadratic settings to demonstrate the applicability of our main result.

Quantitative convergence rates for extended mean field games with volatility control

TL;DR

This work addresses the quantitative convergence of open-loop Nash equilibria for symmetric -player stochastic differential games with interactions through both states and controls, where volatility terms are controlled. It develops a stochastic maximum principle (SMP) framework and extends forward-backward propagation of chaos to the extended mean field setting, proving the well-posedness of conditional McKean–Vlasov FBSDEs via continuation under a displacement monotonicity condition. Under these assumptions, the paper establishes a rigorous convergence rate between the -player Nash equilibria and the mean field equilibrium, both in state trajectories and controls, by constructing a conditional MKV-FBSDE coupling and employing the method of continuation. To illustrate applicability, a linear-quadratic example is analyzed, showing unique existence of the MFE and validating the convergence rate, while avoiding a full closed-loop characterization. The results provide finite- guarantees for extended mean field games with volatility control, with potential applications in finance and economic modeling where control of volatility and control distribution are essential.

Abstract

We investigate the convergence of symmetric stochastic differential games with interactions via control, where the volatility terms of both idiosyncratic and common noises are controlled. We apply the stochastic maximum principle, following the approach of Laurière and Tangpi, to reduce the convergence analysis to the study of forward-backward propagation of chaos. Under the standard monotonicity conditions, we derive quantitative convergence rates for open-loop Nash equilibria of -player stochastic differential games toward the corresponding mean field equilibrium. As a prerequisite, we also establish the well-posedness of the conditional McKean--Vlasov forward-backward stochastic differential equations by the method of continuation. Moreover, we analyze a specific class of linear-quadratic settings to demonstrate the applicability of our main result.
Paper Structure (16 sections, 17 theorems, 181 equations)

This paper contains 16 sections, 17 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Literature review
  3. Structure of this paper
  4. Setting and main results
  5. Notation
  6. Probabilistic setup
  7. The $N$-player game
  8. The mean field game
  9. Main convergence result
  10. Characterization of equilibria via the stochastic maximum principle
  11. The maximum principle for the $N$-player game
  12. The maximum principle for the mean field game
  13. Proof of Theorem \ref{['thm:main']}
  14. Linear-quadratic example
  15. Well-posedness of conditional McKean--Vlasov FBSDEs
  16. ...and 1 more sections

Key Result

Lemma 2.2

Given $X \in L^2(\Omega^N, \mathcal{F}^N, \mathbb{P}^N;\mathbb{R}^k)$, the mapping is almost surely well defined under $\mathbb{P}^0$, and forms a random variable from $(\Omega^0, \mathcal{F}^0, \mathbb{P}^0)$ into $\mathcal{P}_2(\mathbb{R}^k)$ endowed with its Borel $\sigma$-field. The random variable $\mathcal{L}^1(X)$ provides a conditional law of $X$ given $\mathcal{F}^0$. Mor

Theorems & Definitions (44)

  • Remark 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.5
  • Remark 2.7
  • Lemma 2.8
  • Remark 2.9
  • Definition 2.10
  • Lemma 2.11
  • Theorem 2.14
  • ...and 34 more