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

Exact tail asymptotics of the supremum of strongly dependent gaussian processes over a random interval

Let be a positive random variable independent of a real-valued stochastic process . In this paper, we investigate the asymptotic behavior of as u -> a assuming that X is a strongly dependent stationary Gaussian process and has a regularly varying survival function at infinity with index lambda a [0, 1). Under asymptotic restrictions on the correlation function of the process, we show that with some positive finite constant c and function m(center dot) defined in terms of the local behavior of the correlation function and the standard Gaussian distribution