Extremes of Reflecting Gaussian Processes on Discrete Grid

For $\{X(t),t\in G_{\delta }\}$ a centered Gaussian process with stationary increments and a.s. sample paths on a discrete grid $G_{\delta }=\{0,\delta ,2\delta ,\mathrm{.}\mathrm{.}\mathrm{.}\}$, where $\delta >0$, we investigate the stationary reflected process $$Q_{\delta ,X}(t)=\underset{s\in [t,\mathrm{\infty })\cap G_{\delta }}{\sup }(X(s)-X(t)-c(s-t)),t\in G_{\delta }$$ with $c>0$. We derive the exact asymptotics of $$\mathbb{P}\{\underset{t\in [0,T]\cap G_{\delta }}{\sup }{Q}_{\delta ,X}(t)>u\}\text{and}\mathbb{P}\{\underset{t\in [0,T]\cap G_{\delta }}{\inf }{Q}_{\delta ,X}(t)>u\},$$ as $u\to \mathrm{\infty }$, with $T>0$. It appears that $\phi ={\lim }_{u\to \mathrm{\infty }}\frac{{\sigma }^2(u)}{u}$ determines the asymptotics, leading to three qualitatively different scenarios: $\phi =0$, $\phi \in (0,\mathrm{\infty })$ and $\phi =\mathrm{\infty }$.