Derivatives of sup-functionals of fractional Brownian motion evaluated at H= 12

Krzysztof Bisewski; Krzysztof Dȩbicki; Tomasz Rolski

doi:10.1214/22-EJP848

2022 Derivatives of sup-functionals of fractional Brownian motion evaluated at

H=\frac{1}{2}

Krzysztof Bisewski, Krzysztof Dȩbicki, Tomasz Rolski

Author Affiliations +

Electron. J. Probab. 27: 1-35 (2022). DOI: 10.1214/22-EJP848

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 $H=\frac{1}{2}$ 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.

Funding Statement

KB’s research was funded by SNSF Grant 200021-196888. KD and TR were partially supported by NCN Grant No 2018/31/B/ST1/00370 (2019-2022).

Acknowledgments

We would like to thank the anonymous referees for valuable remarks that significantly improved the presentation of the results of this contribution.

Citation

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Krzysztof Bisewski. Krzysztof Dȩbicki. Tomasz Rolski. "Derivatives of sup-functionals of fractional Brownian motion evaluated at $H=\frac{1}{2}$ ." Electron. J. Probab. 27 1 - 35, 2022. https://doi.org/10.1214/22-EJP848