Tools for Malliavin calculus in UMD Banach spaces

TL;DR

This paper develops new tools and results for Malliavin calculus in UMD Banach spaces, facilitating analysis of infinite-dimensional stochastic processes with applications to SPDEs and finance.

Contribution

It introduces novel results such as weak characterizations, a chain rule, and an Ito formula for non-adapted processes in UMD Banach spaces.

Findings

Established weak characterizations of Malliavin derivatives

Proved a chain rule for Lipschitz functions in this setting

Derived an Ito formula for non-adapted processes

Abstract

In this paper we study the Malliavin derivatives and Skorohod integrals for processes taking values in an infinite dimensional space. Such results are motivated by their applications to SPDEs and in particular financial mathematics. Vector-valued Malliavin theory in Banach space E is naturally restricted to spaces E which have the so-called UMD property, which arises in harmonic analysis and stochastic integration theory. We provide several new results and tools for the Malliavin derivatives and Skorohod integrals in an infinite dimensional setting. In particular, we prove weak characterizations, a chain rule for Lipschitz functions, a sufficient condition for pathwise continuity and an Ito formula for non-adapted processes.