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For $\{X(t), t \in G_\delta\}$ a centered Gaussian process with stationaryincrements and a.s. sample paths on a discrete grid$G_\delta=\{0,\delta,2\delta, ...\}$, where $\delta>0$, we investigate thestationary reflected process $$Q_{\delta,X}(t) = \sup\limits_{s\in[t,\infty)\cap G_\delta}\big( X(s)-X(t)-c(s-t)\big), \ t \in G_\delta $$ with$c>0$. We derive the exact asymptotics of $$\mathbb P\{\sup\limits_{t\in[0,T]\cap G_\delta} Q_{\delta,X}(t)>u\} \quad \text{and} \quad \mathbbP\{\inf\limits_{t\in [0,T]\cap G_\delta} Q_{\delta,X}(t)>u\},$$ as$u\to\infty$, with $T>0$. It appears that $\varphi=\lim_{u\to\infty}\frac{\sigma^2(u)}{u}$ determines the asymptotics, leading to threequalitatively different scenarios: $\varphi=0$, $\varphi\in(0,\infty)$ and$\varphi=\infty$.

Extremes of Reflecting Gaussian Processes on Discrete Grid