2013/10/31 by K. Dębicki, Krzysztof Dębicki, Kamil Marcin Kosiński +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Brownian motion #Fractional Brownian motion #Gaussian #Gaussian process #Hurst exponent #Infimum and supremum #Random Matrices and Applications #Real line #Reflected Brownian motion #Stochastic process #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60F15 #msc:60G22 #msc:60G70
paper · pdf · doi:10.1007/s10687-014-0188-7
published as Extremes 17 (2014) 431--446
arxiv created 2014/04/06 · openalex publication_date 2014/07/11 · arxiv updated 2014/09/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let B H (t):t≥0 be a fractional Brownian motion with Hurst parameter H∈ (\frac 12,1) . For the storage process Q_BH(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^\frac 2H-1H) , \mathbb P (infs∈[0,T(u)] Q_BH(s)>u)∼\mathbb P(Q_BH(0)>u), 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.