On the infimum attained by the reflected fractional Brownian motion

Abstract

Let {BH(t):t≥0} be a fractional Brownian motion with Hurst parameter H ∈ ( (Formula presented.) , 1). For the storage process QBH(t) = sup−∞≤s≤t (BH(t) − BH(s) − c(t − s)) we show that, for any T(u)>0 such that T(u) = o(u (Formula presented.) ), (Formula presented.) as u → ∞. 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.