Malliavin derivative of random functions and applications to Lévy driven BSDEs

Abstract

We consider measurable $F: \Omega \times \mathbb{R} ^d \to \mathbb{R} $ where for any $x$ the random variable $F(\cdot , x)$ belongs to the Malliavin Sobolev space $\mathbb{D} _{1,2}$ (with respect to a Lévy process) and provide sufficient conditions on $F$ and $G_1,\ldots ,G_d \in \mathbb{D} _{1,2}$ such that $F(\cdot , G_1,\ldots ,G_d) \in \mathbb{D} _{1,2}.$

The above result is applied to show Malliavin differentiability of solutions to BSDEs (backward stochastic differential equations) driven by Lévy noise where the generator is given by a progressively measurable function $f(\omega ,t,y,z).$