On the infimum attained by the reflected fractional Brownian motion

Abstract

Let $\{B_H(t):t\ge 0\}$ 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 $T(u)=o(u^\frac{2H-1}{H})$, \[\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_H}(s)>u)\sim\mathbb P(Q_{B_H}(0)>u),\quad\text{as}\quad u\to\infty.\] This finding, known in the literature as the strong Piterbarg property, is in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.