Extremes of stationary Gaussian storage models

For the stationary storage process {Q(t), t ≥ 0}, with Q(t)=sups≥tX(s)−X(t)−c(s−t)β,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ Q(t)=\sup _{s\ge t}\left (X(s)-X(t)-c(s-t)^{\beta }\right ),$\end{document} where {X(t), t ≥ 0} is a centered Gaussian process with stationary increments, c > 0 and β > 0 is chosen such that Q(t) is finite a.s., we derive exact asymptotics of ℙsupt∈[0,Tu]Q(t)>u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {P}\left (\sup _{t\in [0,T_{u}]} Q(t)>u \right )$\end{document} and ℙinft∈[0,Tu]Q(t)>u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {P}\left (\inf _{t\in [0,T_{u}]} Q(t)>u \right )$\end{document}, as u→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u\rightarrow \infty $\end{document}. As a by-product we find conditions under which strong Piterbarg property holds.