[Paper Review] Propagation of Chaos of Forward-Backward Stochastic Differential Equations with Graphon Interactions

Erhan Bayraktar, Ruoyu Wu+1 more|arXiv (Cornell University)|Feb 16, 2022

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes propagation of chaos for forward-backward stochastic differential equations (FBSDEs) with graphon interactions, proving convergence of n-player game Nash equilibria to the solution of a graphon mean field game. Using contraction mapping and continuation methods under monotonicity conditions, it demonstrates existence, uniqueness, and stability of solutions, enabling the first general convergence result for graphon mean field games beyond linear-quadratic models.

ABSTRACT

In this paper, we study graphon mean field games using a system of forward-backward stochastic differential equations. We establish the existence and uniqueness of solutions under two different assumptions and prove the stability with respect to the interacting graphons which are necessary to show propagation of chaos results. As an application of propagation of chaos, we prove the convergence of n-player game Nash equilibrium for a general model, which is new in the theory of graphon mean field games.

Motivation & Objective

• To establish existence and uniqueness of solutions for a limiting system of forward-backward stochastic differential equations (FBSDEs) with graphon interactions.
• To prove stability of solutions with respect to perturbations in the graphon interaction kernel.
• To demonstrate propagation of chaos for a system of n interacting agents governed by FBSDEs with graphon interactions.
• To show convergence of n-player game Nash equilibria to the solution of the graphon mean field game in a general model setting.
• To extend the theory of mean field games to heterogeneous, graphon-based interaction structures beyond linear-quadratic models.

Proposed method

• Formulates a system of n-coupled FBSDEs where interaction strengths are determined by a graphon sequence $ G_n $, approximating a limiting graphon $ G $.
• Introduces a limiting FBSDE system indexed by $ \lambda \in [0,1] $, where interactions are integrated over the graphon kernel $ G(\lambda,\kappa) $ and the law of $ X_t^\kappa $.
• Uses the method of continuation and contraction mapping to prove existence and uniqueness of solutions under two monotonicity assumptions, generalizing results from classical FBSDE theory.
• Establishes measurability of the map $ \lambda \mapsto \mathcal{L}(X_t^\lambda) $ by considering a common Brownian motion for all $ \lambda $, preserving marginal laws.
• Proves stability of solutions with respect to graphon perturbations using uniform estimates under the same monotonicity conditions.
• Applies propagation of chaos to show that the empirical distribution of the n-player system converges to the solution of the limiting graphon FBSDE system.

Experimental results

Research questions

• RQ1 Under what conditions does a system of n interacting agents governed by FBSDEs with graphon interactions exhibit propagation of chaos?
• RQ2 How can existence and uniqueness of solutions be established for the limiting graphon FBSDE system with non-symmetric, heterogeneous interactions?
• RQ3 What conditions ensure stability of the solution map with respect to changes in the graphon interaction kernel?
• RQ4 Can the Nash equilibrium of the finite n-player game converge to the solution of the graphon mean field game in a general, non-linear model?
• RQ5 How can the law of the process $ X^\lambda $ be shown to be measurable in $ \lambda $, given the dependence on the joint law of the system?

Key findings

• The paper proves existence and uniqueness of solutions to the limiting graphon FBSDE system under two standard monotonicity conditions, extending classical FBSDE results to graphon interactions.
• Stability of solutions with respect to graphon perturbations is established, which is essential for proving propagation of chaos.
• Propagation of chaos is proven: the finite n-player system converges in law to the limiting graphon FBSDE system as $ n \to \infty $.
• The convergence of n-player game Nash equilibria to the graphon mean field game solution is established for a general model, a novel result beyond linear-quadratic settings.
• Measurability of the map $ \lambda \mapsto \mathcal{L}(X_t^\lambda) $ is rigorously proven using a common Brownian motion representation, enabling the definition of the interaction integral in the limiting system.
• The results are obtained via contraction mapping and continuation methods, with uniform estimates ensuring convergence and stability under graphon convergence $ G_n \to G $.