English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We provide upper and lower bounds for the mean ${\mathscr M}(H)$ of$\sup_{t\geqslant 0} \{B_H(t) - t\}$, 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 anumerical 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 astrongly similar shape. We also derive a new upper bound for the mean of$\sup_{t\in[0,1]} B_H(t)$, $H\in(0,\tfrac{1}{2}]$, which is tight around$H=\tfrac{1}{2}$.

Bounds for expected supremum of fractional Brownian motion with drift