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

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= 12 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.