Backward propagation of chaos

TL;DR

This paper establishes a theory for the propagation of chaos in backward stochastic differential equations with fixed terminal conditions, providing convergence rates and concentration inequalities for interacting particle systems.

Contribution

It introduces a novel propagation of chaos framework for backward systems with fixed terminal states, including quantitative convergence estimates and applications to PDEs.

Findings

Proves propagation of chaos for backward particle systems.

Provides explicit convergence rates in Wasserstein distance.

Derives convergence results for solutions of semilinear PDEs.

Abstract

This paper develops a theory of propagation of chaos for a system of weakly interacting particles whose terminal configuration is fixed as opposed to the initial configuration as customary. Such systems are modeled by backward stochastic differential equations. Under standard assumptions on the coefficients of the equations, we prove propagation of chaos results and quantitative estimates on the rate of convergence in Wasserstein distance of the empirical measure of the interacting system to the law of a McKean-Vlasov type equation. These results are accompanied by non-asymptotic concentration inequalities. As an application, we derive rate of convergence results for solutions of second order semilinear partial differential equations to the solution of a partial differential written on an infinite dimensional space.