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Strong solutions of mean-field FBSDEs and their applications to multi-population mean-field games

Kihun Nam, Yunxi Xu

TL;DR

The paper addresses the existence of strong solutions for mean-field forward–backward SDEs with irregular coefficients and leverages these results to establish Nash equilibria in multi-population mean-field games. It develops a fixed-point approach on the law flow $\mu_t=\mathcal{L}^H(X_t)$, employing a Girsanov transform to solve decoupled FBSDEs for fixed $\mu$ and then using Schauder’s fixed-point theorem to obtain a fixed point that yields a strong solution to the coupled system. The work accommodates discontinuities in the forward dynamics and non-Lipschitz dependence on the mean-field term, and provides both weak- and strong-form formulations for MP-MFG, including a Hamiltonian system with a measurable selector for optimal controls and a localization argument to handle linear growth. It also clarifies conditions under which uniqueness holds in the decoupled case and illustrates a one-dimensional example where a Nash equilibrium can be explicitly characterized. Overall, the results give a rigorous probabilistic framework for strong solvability and equilibrium analysis in large, multi-population systems with irregular mean-field interactions.

Abstract

We study the existence of strong solutions for mean-field forward-backward stochastic differential equations (FBSDEs) with measurable coefficients and their implication on the Nash equilibrium of a multi-population mean-field game. More specifically, we allow the coefficients to be discontinuous in the forward process and non-Lipschitz continuous concerning their time-sectional distribution. Using the Pontryagin stochastic maximum principle and the martingale approach, we apply our existence result to a multi-population mean-field game (MPMFG) model where the interacting agents in the system are grouped into multiple populations. Each population shares the same objective function, and we take changes in population sizes into consideration.

Strong solutions of mean-field FBSDEs and their applications to multi-population mean-field games

TL;DR

The paper addresses the existence of strong solutions for mean-field forward–backward SDEs with irregular coefficients and leverages these results to establish Nash equilibria in multi-population mean-field games. It develops a fixed-point approach on the law flow , employing a Girsanov transform to solve decoupled FBSDEs for fixed and then using Schauder’s fixed-point theorem to obtain a fixed point that yields a strong solution to the coupled system. The work accommodates discontinuities in the forward dynamics and non-Lipschitz dependence on the mean-field term, and provides both weak- and strong-form formulations for MP-MFG, including a Hamiltonian system with a measurable selector for optimal controls and a localization argument to handle linear growth. It also clarifies conditions under which uniqueness holds in the decoupled case and illustrates a one-dimensional example where a Nash equilibrium can be explicitly characterized. Overall, the results give a rigorous probabilistic framework for strong solvability and equilibrium analysis in large, multi-population systems with irregular mean-field interactions.

Abstract

We study the existence of strong solutions for mean-field forward-backward stochastic differential equations (FBSDEs) with measurable coefficients and their implication on the Nash equilibrium of a multi-population mean-field game. More specifically, we allow the coefficients to be discontinuous in the forward process and non-Lipschitz continuous concerning their time-sectional distribution. Using the Pontryagin stochastic maximum principle and the martingale approach, we apply our existence result to a multi-population mean-field game (MPMFG) model where the interacting agents in the system are grouped into multiple populations. Each population shares the same objective function, and we take changes in population sizes into consideration.
Paper Structure (10 sections, 12 theorems, 125 equations)

This paper contains 10 sections, 12 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Wellposedness of mean-field FBSDE
  4. Existence of a Solution
  5. Uniqueness of a Solution
  6. Mean-field FBSDE and multi-population mean-field game
  7. Model setup
  8. Strong solvability of multi-population mean-field games in weak formulation
  9. Multi-population mean-field game and corresponding Hamiltonian system
  10. Measure Change of FBSDE

Key Result

Lemma 3.2

For any given $\mu\in\mathcal{C}\left([0,T];\mathcal{P}^H_1(\mathbb R^m)\right)$, the FBSDE has a unique strong solution $(X^\mu,Y^\mu,Z^\mu)\in\mathbb H^2\times\mathbb H^2\times\mathbb H^2$.

Theorems & Definitions (35)

  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • proof
  • Remark 3.5
  • Proposition 3.6
  • proof
  • Definition 3.7
  • ...and 25 more