Extremes of stationary Gasussian storage models

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 $\mathbb{P}({\sup }_{t\in [0,T_u]}Q(t)>u)$ and $\mathbb{P}({\inf }_{t\in [0,T_u]}Q(t)>u)$, as $u\to \mathrm{\infty }$. As a by-product we find conditions under which strong Piterbarg property holds.