---
abstract: 'In this paper we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. Our results rely on the interpretation of the Malliavin derivative as a Gâteaux derivative in the directions of the Cameron-Martin space. Incidentally, we provide a new formulation for the characterization of the Malliavin-Sobolev type spaces $\D^{1,p}$.'
author:
- 'Thibaut Mastrolia [^1]'
- 'Dylan Possamaï[^2]'
- 'Anthony Réveillac[^3]'
title: On the Malliavin differentiability of BSDEs
---

[*AMS 2010 subject classification:* Primary: 60H10; Secondary: 60H07. ]{}

Introduction
============

Backward Stochastic Differential Equations (BSDEs) have been studied extensively in the last two decades as they naturally arise in the context of stochastic control problems (for instance in Finance see [@EPQ]), and as they provide a probabilistic representation for solution to semi-linear parabolic PDEs, via a non-linear Feynman-Kac formula (see [@pardouxpeng]). Before going further let us recall that this class of equations has been introduced in [@Bismut_78; @Pardoux_Peng90; @pardouxpeng] and that a BSDE can be formulated as: $$\label{eq:BSDE}
Y_t = \xi +\int_t^T f(s,Y_s,Z_s)ds -\int_t^T Z_s dW_s, \quad t\in [0,T],$$ where $T$ is a fixed positive number, $W:=(W_t)_{t\in [0,T]}$ is a one-dimensional Brownian motion defined on a probability space $(\Omega,\mathcal{F}_T,\P)$ with natural filtration $(\mathcal{F}_t)_{t\in [0,T]}$. 