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

Krzysztof Bisewski (), Krzysztof Dȩbicki () and Tomasz Rolski ()
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Krzysztof Bisewski: University of Lausanne
Krzysztof Dȩbicki: University of Wrocław
Tomasz Rolski: University of Wrocław

Queueing Systems: Theory and Applications, 2022, vol. 102, issue 1, No 4, 53-68

Abstract: Abstract The H-derivative of the expected supremum of fractional Brownian motion $$\{B_H(t),t\in {\mathbb {R}}_+\}$$ { B H ( t ) , t ∈ R + } with drift $$a\in {\mathbb {R}}$$ a ∈ R over time interval [0, T] $$\begin{aligned} \frac{\partial }{\partial H} {\mathbb {E}}\Big (\sup _{t\in [0,T]} B_H(t) - at\Big ) \end{aligned}$$ ∂ ∂ H E ( sup t ∈ [ 0 , T ] B H ( t ) - a t ) at $$H=1$$ H = 1 is found. This formula depends on the quantity $${\mathscr {I}}$$ I , which has a probabilistic form. The numerical value of $${\mathscr {I}}$$ I is unknown; however, Monte Carlo experiments suggest $${\mathscr {I}}\approx 0.95$$ I ≈ 0.95 . As a by-product we establish a weak limit theorem in C[0, 1] for the fractional Brownian bridge, as $$H\uparrow 1$$ H ↑ 1 .

Keywords: fractional Brownian motion; expected supremum; H-derivative; 60G17; 60G22; 60G70 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11134-022-09859-3

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