On the infimum attained by the reflected fractional Brownian motion. (English) Zbl 1306.60037
Summary: Let \(\{B_{H}(t):t\geq0\}\) be a fractional Brownian motion with Hurst parameter \(H\in \left(\frac {1}{2},1\right)\). For the storage process \(Q_{B_{H}}(t)=\sup _{-\infty \leq s\leq t}\left (B_{H}(t)-B_{H}(s)-c(t-s)\right)\) we show that, for any \(T(u)>0\) such that \(T(u)=o\left(u^{\frac {2H-1}{H}}\right)\),
\[
\operatorname{P} \left(\inf_{s\in[0,T(u)]} Q_{B_{H}}(s)>u\right)\sim \operatorname{P}(Q_{B_{H}}(0)>u),
\]
as \(u\to \infty \). This finding, known in the literature as the strong Piterbarg property, goes in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60G70 | Extreme value theory; extremal stochastic processes |
| 60F15 | Strong limit theorems |
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