On the uniqueness result for the BSDE with continuous coefficient

TL;DR

This paper establishes uniqueness results for one-dimensional BSDEs with Lipschitz continuous in y and merely continuous in z coefficients, under certain conditions on the terminal condition's Malliavin derivative.

Contribution

It provides new uniqueness theorems for BSDEs with continuous in z coefficients, extending previous results to cases with quadratic and linear growth.

Findings

Proves uniqueness for BSDEs with quadratic growth in z.

Proves uniqueness for BSDEs with linear growth in z.

Requires terminal condition with bounded Malliavin derivative.

Abstract

In this paper, we study one-dimensional backward stochastic differential equation (BSDE, for short), whose coefficient $f$ is Lipschitz in $y$ but only continuous in $z$. In addition, if the terminal condition $\xi$ has bounded Malliavin derivative, we prove some uniqueness results for the BSDE with quadratic and linear growth in $z$, respectively.