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

Christel Geiss; Alexander Steinicke

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

Probability 2016-09-26 v4

Authors: Christel Geiss , Alexander Steinicke

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 L\'evy 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 L\'evy noise where the generator is given by a progressively measurable function $f(\omega,t,y,z).$

Keywords

Malliavin derivative random functions Levy process backward stochastic differential equations Malliavin Sobolev space

Cite

@article{arxiv.1404.4477,
  title  = {Malliavin derivative of random functions and applications to L\'evy driven BSDEs},
  author = {Christel Geiss and Alexander Steinicke},
  journal= {arXiv preprint arXiv:1404.4477},
  year   = {2016}
}

Comments

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

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

Abstract

Keywords

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