Extremes of stationary Gasussian storage models

For the stationary storage process $\{Q(t), t\ge0\}$, 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 $u\rightarrow\infty$. As a by-product we find conditions under which strong Piterbarg property holds.