Extremes of stationary Gasussian storage models

Abstract

For the stationary storage process $\{Q(t),t\ge 0\}$, with $Q(t)={\sup }_{s\ge t}\left(X(s)-X(t)-c(s-t{)}^{\beta }\right),$ where $\{X(t),t\ge 0\}$ is a centered Gaussian process with stationary increments, $c>0$ and $\beta >0$ 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 $u\to \infty$. As a by-product we find conditions under which strong Piterbarg property holds.

23 pages