Time discretization of Quadratic Forward-Backward SDEs with singular drifts

TL;DR

This paper studies the convergence rate of time discretization schemes for quadratic backward SDEs with singular drifts, revealing how noise regularization improves numerical approximation accuracy.

Contribution

It introduces a novel analysis connecting noise regularization effects to enhanced convergence rates in discretizing singular forward-backward SDEs.

Findings

Achieves an error rate close to 1/2 for explicit schemes.

Extends Zhang's L^2-time regularity to singular SDEs.

Demonstrates noise regularization improves numerical convergence.

Abstract

We investigate the convergence rate for the time discretization of a class of quadratic backward SDEs -- potentially involving path-dependent terminal values -- when coupled with non-standard Lipschitz-type forward SDEs. In our review of the explicit time-discretization schemes in the spirit of Pag\`es \& Sagna (see \cite{PaSa18}), we achieve an error control close to $\frac{1}{2}$, even under the modest assumptions considered in this work (see \cite{ChaRichou16}, for comparison). A central element of our approach is a thorough re-examination of Zhang's $L^2\text{-time regularity}$ of the martingale integrand $Z$ which follows from an extension of the first-order variational regularity for this class of singular forward-backward SDEs with non-uniform Cauchy-Lipschitz drivers. This is complemented by the recently introduced caracterisation of stochastic processes of {\it bounded mean oscillation} (abbreviated as $\bmo$) by K. L\^e (see \cite{Le22}) which we used to derive an $L^p\text{-version}$ of the strong approximation of SDEs with singular drifts from Dareiotis \& Gerencs\'er (see \cite{DaGe20}). As such, this study addresses a crucial gap in the numerical analysis of forward-backward SDEs (FBSDEs). To our knowledge, for the first time, the impact of regularization by noise on Euler-Maruyama numerical schemes for singular forward SDEs has been successfully transferred to enhance the convergence rate of the discrete time approximations for solutions to backward SDEs.