Extremes of

For a given centered Gaussian process with stationary increments X(t),t ≥ 0 and c > 0, let Wγ(t) = X(t) − ct − γinf0 ≤ s ≤ t(X(s) − cs),t ≥ 0 denote the γ-reflected process, where γ ∈ (0,1). This process is important for both queueing and risk theory. In this contribution we are concerned with the asymptotics, as u → ∞, of ℙ(sup0≤t ≤ T Wγ(t)>u), t ∈ (o,∞]. Moreover, we investigate the approximations of first and last passage times for given large threshold u. We apply our findings to the cases with X being the multiplex fractional Brownian motion and the Gaussian integrated process. As a by-product we derive an extension of Piterbarg inequality for threshold-dependent random fields