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

• Main Authors: Bisewski, Krzysztof, Dębicki, Krzysztof, Rolski, Tomasz
• Format: Journal Article
• Language: English
• Published: 17-10-2021
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.2110.08788

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.