Chaotic extensions and the lent particle method for Brownian motion

TL;DR

This paper extends the lent particle method from Poisson spaces to Wiener spaces, providing a new formula for Malliavin derivatives of functionals using chaos extensions and stationary processes.

Contribution

It introduces a novel lent particle formula for Wiener space, enhancing the calculation of Malliavin derivatives for functionals of Brownian motion.

Findings

Established a lent particle formula for Wiener space

Simplified the computation of Malliavin derivatives

Connected chaos extensions with stationary processes

Abstract

In previous works, we have developed a new Malliavin calculus on the Poisson space based on the lent particle formula. The aim of this work is to prove that, on the Wiener space for the standard Ornstein-Uhlenbeck structure, we also have such a formula which permits to calculate easily and intuitively the Malliavin derivative of a functional. Our approach uses chaos extensions associated to stationary processes of rotations of normal martingales.