Derivatives of sup-functionals of fractional Brownian motion evaluated at H=1/2

TL;DR

This paper derives explicit formulas for the derivatives of various sup-functionals of fractional Brownian motion at H=1/2, enhancing understanding of their sensitivity to the Hurst parameter.

Contribution

It introduces a novel approach to compute derivatives of sup-functionals of fractional Brownian motion at H=1/2 using fractional stable fields and Paley-Wiener-Zygmund representation.

Findings

Explicit formulas for derivatives at H=1/2 are established.

The concept of derivatives of fractional stable fields is developed.

A new representation of fractional Brownian motion is proposed.

Abstract

We consider a family of sup-functionals of (drifted) fractional Brownian motion with Hurst parameter $H\in(0,1)$. This family includes, but is not limited to: expected value of the supremum, expected workload, Wills functional, and Piterbarg-Pickands constant. Explicit formulas for the derivatives of these functionals as functions of Hurst parameter evaluated at $H=\tfrac{1}{2}$ are established. In order to derive these formulas, we develop the concept of derivatives of fractional $\alpha$-stable fields introduced by Stoev \& Taqqu (2004) and propose Paley-Wiener-Zygmund representation of fractional Brownian motion.