Extremes of threshold-dependent Gaussian processes

In this contribution we are concerned with the asymptotic behaviour, as u→∞, of P{supt∈[0,T]Xu(t)>u}, where Xu(t),t∈[0,T],u>0 is a family of centered Gaussian processes with continuous trajectories. A key application of our findings concerns P{supt∈[0,T](X(t)+g(t))>u}, as u→∞, for X a centered Gaussian process and g some measurable trend function. Further applications include the approximation of both the ruin time and the ruin probability of the Brownian motion risk model with constant force of interest.