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

2021/10/17 by Krzysztof Bisewski, Bisewski, Krzysztof, Krzysztof Dȩbicki +3
Economics, Econometrics and Finance · #60G17 #60G22 #60G70 #FOS: Mathematics #Financial Markets and Investment Strategies #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2110.08788

openalex publication_date 2021/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a family of sup-functionals of (drifted) fractional Brownian motion with Hurst parameter H∈(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=\tfrac12 are established. In order to derive these formulas, we develop the concept of derivatives of fractional α-stable fields introduced by Stoev & Taqqu (2004) and propose Paley-Wiener-Zygmund representation of fractional Brownian motion.

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