Gaussian approximation on the Skorokhod space via Malliavin calculus and regularization

TL;DR

This paper develops a Gaussian approximation method in the Skorokhod space using Malliavin calculus, introducing a new operator and bounds for Banach-valued random elements.

Contribution

It introduces a novel carré du champ operator for Banach-valued elements and provides bounds for Gaussian approximation in the Skorokhod space.

Findings

Bound the bounded Lipschitz distance using the new operator.

Derived bounds for Banach-valued multiple integrals.

Extended recent Hilbert space results to Banach spaces.

Abstract

We introduce a carr\'e du champ operator for Banach-valued random elements, taking values in the projective tensor product, and use it to control the bounded Lipschitz distance between a Malliavin-smooth random element satisfying mild regularity assumptions and a Radon Gaussian taking values in the Skorokhod space equipped with the uniform topology. In the case where the random element is a Banach-valued multiple integral, the carr\'e du champ expression is further bounded by norms of the contracted integral kernel. The main technical tool is an integration by parts formula, which might be of independent interest. As a by-product, we recover a bound obtained recently by D\"uker and Zoubouloglou in the Hilbert space setting and complement it by providing contraction bounds.