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

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

33 pages, 0 figures