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 ( t ) , t ∈ R + } with drift a ∈ R over time interval [0,  T ] ∂ ∂ H E ( sup t ∈ [ 0 , T ] B H ( t ) - a t ) 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 ≈ 0.95 . As a by-product we establish a weak limit theorem in C [0, 1] for the fractional Brownian bridge, as H ↑ 1 .