Convergence rate for random walk approximations of mean field BSDEs

TL;DR

This paper investigates the convergence rate of random walk approximations for mean field BSDEs using a novel freezing technique, extending classical results to the mean field setting with new phenomena and technical challenges.

Contribution

It introduces a new approach avoiding particle methods, extending convergence results to mean field BSDEs, and develops techniques to handle singularities and generator behaviors.

Findings

Achieves polynomial approximation rates up to a logarithmic factor.

Introduces a modified H"older continuity concept for singular terminal conditions.

Provides convergence rates for the integrated gradient process in Lorentz spaces.

Abstract

We study the rate of convergence w.r.t.~a Wasserstein type distance for random walk approximations of mean field BSDEs. Our method does not use the particle method but instead a freezing technique. We extend results by Briand, Ch. Geiss, S. Geiss, and Labart [Bernoulli, 27(2) 2021] about the rate of convergence of a Donsker-type theorem for BSDEs from the classical setting to the mean field setting. In this connection the mean field setting leads to new phenomena and requires new techniques that should be of independent interest: The H\"older continuous terminal condition causes a singularity in time of the generator when seen as a generator in the non-mean field setting. To handle this singularity we introduce a concept of modified H\"older continuity by which we are able to achieve, up to a logarithmic term, the same polynomial approximation rates as in the classical non-mean field setting (in fact, already when approximating the Brownian motion itself a logarithmic term is necessary). Moreover, the exploited freezing technique of the mean field terms yields to the problem to handle the quantitative behavior of several different generators. Using BMO-techniques we obtain the rate of convergence for the integrated gradient process in the scale of Lorentz (type) spaces of exponential type.