Sharp propagation of chaos for mean-field backward stochastic differential equations

Abstract

We study quantitative propagation of chaos for mean-field forward-backward stochastic differential equations with interactions through the empirical measures of the forward states, backward values, and diagonal martingale integrands. Our main contribution is to introduce an intermediate-particle comparison method that transfers forward propagation of chaos estimates to the backward system while controlling the full matrix of martingale integrands. Using this method, we establish sharp rates for the backward values and diagonal martingale integrands, with order $m^2/n^2$ in squared Wasserstein distance and order $n^{-1}$ in smooth weak metrics for each fixed $m$. Explicit examples provide matching lower bounds, showing the optimality of these rates. We also establish coupling estimates, including a squared Wasserstein bound of order $m/n$ with dimension-independent constants under suitable assumptions.