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Derivatives of sup-functionals of fractional Brownian motion evaluated at H=1/2
Krzysztof Bisewski, Krzysztof Dębicki, Tomasz Rolski · 2021-10-17 · via math.PR updates on arXiv.org

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 $α$-stable fields introduced by Stoev \& Taqqu (2004) and propose Paley-Wiener-Zygmund representation of fractional Brownian motion.

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