Derivative of the expected supremum of fractional Brownian motion at $H=1$

The H-derivative of the expected supremum of fractional Brownian motion $\{B_H\left(t\right),t\in R_{+}\}$ with drift $a\in R$ over time interval [0, T] $\begin{array}{c}\frac{\partial }{\partial H}E(\underset{t\in [0,T]}{sup}{B}_H\left(t\right)-at)\end{array}$ at $H=1$ is found. This formula depends on the quantity $I$ , which has a probabilistic form. The numerical value of $I$ is unknown; however, Monte Carlo experiments suggest $I\approx 0.95$ . As a by-product we establish a weak limit theorem in C[0, 1] for the fractional Brownian bridge, as $H\uparrow 1$ .