BV-regularity for the Malliavin Derivative of the Maximum of the Wiener Process

TL;DR

This paper investigates the Malliavin derivative of the maximum of a Wiener process, showing it admits a measure with finite total variation that is singular with respect to the Wiener measure, revealing new regularity properties.

Contribution

It establishes the second Malliavin derivative measure for the Wiener process maximum and characterizes its singularity and finiteness properties.

Findings

The maximum of a Wiener process admits a second Malliavin derivative measure.

This measure has finite total variation.

The measure is singular with respect to the Wiener measure.

Abstract

We prove that, on the classical Wiener space, the random variable $\sup_{0\le t \le T} W_t$ admits a measure as second Malliavin derivative, whose total variation measure is finite and singular w.r.t.\ the Wiener measure.