Malliavin derivative of random functions and applications to L\'evy driven BSDEs

摘要

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 L\'evy process) and provide sufficientconditions 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 solutionsto BSDEs (backward stochastic differential equations) driven by L\'evy noisewhere the generator is given by a progressively measurable function$f(\omega,t,y,z).$