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 is an element of R+} with drift alpha is an element of R over time interval [0, T] partial derivative/partial derivative H E (sup(t is an element of[0,T]) B-H(t) - at) at H = 1 is found. This formula depends on the quantity J, which has a probabilistic form. The numerical value of J is unknown; however, Monte Carlo experiments suggest J approximate to 0.95. As a by-product we establish a weak limit theorem in C[0, 1] for the fractional Brownian bridge, as H up arrow 1.