Extremes of stationary Gaussian storage models

Abstract

For the stationary storage process {Q(t), t ≥ 0}, with $Q(t)=\sup _{s\ge t}\left (X(s)-X(t)-c(s-t)^{\beta }\right ),$ 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 $\mathbb {P}\left (\sup _{t\in [0,T_{u}]} Q(t)>u \right )$ and $\mathbb {P}\left (\inf _{t\in [0,T_{u}]} Q(t)>u \right )$, as $u\rightarrow \infty$. As a by-product we find conditions under which strong Piterbarg property holds.