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

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