EXTREMES OF γ-REFLECTED GAUSSIAN PROCESSES WITH STATIONARY INCREMENTS

For a given centered Gaussian process with stationary increments X(t), t >= 0 and c > 0, let W-gamma.(t) = X(t) - ct - gamma inf(0 <= s <= t) (X(s) - cs), t >= 0 denote the gamma-reflected process, where gamma is an element of(0, 1). This process is important for both queueing and risk theory. In this contribution we are concerned with the asymptotics, as u -> infinity, of P ( sup(0 <= t <= T) W-gamma(t) > u), T is an element of(0,infinity]. 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.