Differentiability of quadratic forward-backward SDEs with rough drift

TL;DR

This paper investigates the differentiability of quadratic forward-backward stochastic differential equations with non-Lipschitz drift and nonlinear drivers, providing derivative representations and analyzing numerical approximation convergence.

Contribution

It establishes the Malliavin and classical derivatives for QFBSDEs with rough drift and nonlinear drivers, extending previous results to more general conditions.

Findings

Proves differentiability of QFBSDEs with non-Lipschitz drift.

Provides representations for derivatives of the system.

Shows convergence rate of numerical approximation matches previous Lipschitz cases.

Abstract

In this paper, we consider quadratic forward-backward SDEs (QFBSDEs), for {which} the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system {possesses} nonlinearity of type $f(|y|)|z|^2,$ where $f$ is any locally integrable function. We prove both the Malliavin and classical derivative of the QFBSDE and provide representations of these processes. We study a numerical approximation of this system in the sense of \cite{ImkDosReis} in which the authors assume that the drift is Lipschitz and the driver of the BSDE is quadratic in the traditional sense (i.e., $f$ is a positive constant). We show that the rate of convergence is the same as in \cite{ImkDosReis}