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

Abstract

The H-derivative of the expected supremum of fractional Brownian motion {B_H(t),tin {mathbb {R}}_+} with drift ain {mathbb {R}} over time interval [0, T] ∂∂HE(supt∈[0,T]BH(t)-at)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\partial }{\\partial H} {\\mathbb {E}}\\Big (\\sup _{t\\in [0,T]} B_H(t) - at\\Big ) \\end{aligned}$$\\end{document}at H=1 is found. This formula depends on the quantity {mathscr {I}}, which has a probabilistic form. The numerical value of {mathscr {I}} is unknown; however, Monte Carlo experiments suggest {mathscr {I}}approx 0.95. As a by-product we establish a weak limit theorem in C[0, 1] for the fractional Brownian bridge, as Huparrow 1.