Extremes of reflecting Gaussian processes on discrete grid

Abstract

For {X(t),t∈Gδ} a centered Gaussian process with variance σ2 and stationary increments on a discrete grid Gδ={0,δ,2δ,...}, where δ>0, we investigate the stationary reflected processQδ,X(t)=sups∈[t,∞)∩Gδ⁡(X(s)−X(t)−c(s−t)),t∈Gδ with c>0. We derive the exact asymptotics of P(supt∈[0,T]∩Gδ⁡Qδ,X(t)>u) and P(inft∈[0,T]∩Gδ⁡Qδ,X(t)>u), as u→∞, with T>0. It appears that φ=limu→∞⁡σ2(u)u determines the asymptotics, leading to three qualitatively different scenarios: φ=0, φ∈(0,∞) and φ=∞.