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Bounds for expected supremum of fractional Brownian motion with drift

Part of: Stochastic processes Computer system organization

Published online by Cambridge University Press:  23 June 2021

Krzysztof Bisewski Open the ORCID record for Krzysztof Bisewski [Opens in a new window] ,
Krzysztof Dębicki  and
Michel Mandjes
Krzysztof Bisewski*
Affiliation:
Université de Lausanne
Krzysztof Dębicki*
Affiliation:
Wrocław University
Michel Mandjes*
Affiliation:
University of Amsterdam
*
*Postal address: Quartier UNIL-Chamberonne, Bâtiment Extranef, 1015 Lausanne, Switzerland. Email address: [email protected]
**Postal address: pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
***Postal address: Science Park 904, 1098 XH Amsterdam, The Netherlands.
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Abstract

We provide upper and lower bounds for the mean $\mathscr{M}(H)$ of $\sup_{t\geq 0} \{B_H(t) - t\}$ , 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 $H=\frac{1}{2}$ .

Keywords

Fractional Brownian motionextreme valuebounds

MSC classification

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 60G15: Gaussian processes 68M20: Performance evaluation; queueing; scheduling
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 411 - 427
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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