Abstract
We consider a family of sup-functionals of (drifted) fractional Brownian motion with Hurst parameter . 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 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
Krzysztof Bisewski. Krzysztof Dȩbicki. Tomasz Rolski. "Derivatives of sup-functionals of fractional Brownian motion evaluated at ." Electron. J. Probab. 27 1 - 35, 2022. https://doi.org/10.1214/22-EJP848
Information