New and refined bounds for expected maxima of fractional Brownian motion

TL;DR

This paper derives new bounds for the expected maximum of fractional Brownian motion with Hurst parameter H in (0,1/2), improving previous estimates and providing tighter bounds for Pickands' constant.

Contribution

It introduces refined bounds for the expected maximum of fractional Brownian motion and improves existing upper bounds for Pickands' constant.

Findings

New upper and lower bounds for the expected maximum of $B^H$ over [0,1].

Enhanced upper bounds for the expectation of the maximum of $B^H$ over [0,1].

New upper bounds for Pickands' constant.

Abstract

For the fractional Brownian motion $B^H$ with the Hurst parameter value $H$ in (0,1/2), we derive new upper and lower bounds for the difference between the expectations of the maximum of $B^H$ over [0,1] and the maximum of $B^H$ over the discrete set of values $ in^{-1},$ $i=1,\ldots, n.$ We use these results to improve our earlier upper bounds for the expectation of the maximum of $B^H$ over $[0,1]$ and derive new upper bounds for Pickands' constant.