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 $F(\cdot ,x)$ belongs for any $x$ to the Malliavin Sobolev space ${\mathbb{D}}_{1,2}$ (with respect to a Lévy process) and provide sufficient conditions on $F$ and $G_1,\dots ,G_d\in {\mathbb{D}}_{1,2}$ such that $F(\cdot ,G_1,\dots ,G_d)\in {\mathbb{D}}_{1,2}.$ 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 $f(\omega ,t,y,z).$

41 pages. In Theorem 3.12 (iii) and Assumption ($A_f$) e) the local Lipschitz condition on the Malliavin derivative of the generator has been weakened by introducing a map $\rho$ which determines the degree of a function's uniform continuity. One step in the proof of Theorem 3.12 has been corrected assuming slightly stronger integrability conditions in the assumptions of the Theorem