Bounds for expected supremum of fractional Brownian motion with drift

Abstract

We provide upper and lower bounds for the mean $\mathcal{M}(H)$ of ${\sup }_{t⩾0}\{B_H(t)-t\}$, with $B_H(\cdot )$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in (0,1)$. We find bounds in (semi-)closed-form, distinguishing between $H\in (0,\frac{1}{2}]$ and $H\in [\frac{1}{2},1)$, where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For $H\in (0,\frac{1}{2}]$, the ratio between the upper and lower bound is bounded, whereas for $H\in [\frac{1}{2},1)$ the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of ${\sup }_{t\in [0,1]}{B}_H(t)$, $H\in (0,\frac{1}{2}]$, which is tight around $H=\frac{1}{2}$.

16 pages, 3 figures