Abstract
Let {B H (t):t≥0} be a fractional Brownian motion with Hurst parameter \(H\in (\frac {1}{2},1)\). For the storage process \(Q_{B_{H}}(t)=\sup _{-\infty \le s\le t}\) \(\left (B_{H}(t)-B_{H}(s)-c(t-s)\right )\) we show that, for any T(u)>0 such that \(T(u)=o(u^{\frac {2H-1}{H}})\),
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.
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Dębicki, K., Kosiński, K.M. On the infimum attained by the reflected fractional Brownian motion. Extremes 17, 431–446 (2014). https://doi.org/10.1007/s10687-014-0188-7
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DOI: https://doi.org/10.1007/s10687-014-0188-7