Extremes of threshold-dependent Gaussian processes

Abstract

In this contribution we are concerned with the asymptotic behaviour as $u\to \infty$ of $\mathbb{P}\{\sup_{t\in [0,T]} X_u(t)> u\}$, where $X_u(t),t\in [0,T],u>0$ is a family of centered Gaussian processes with continuous trajectories. A key application of our findings concerns $\mathbb{P}\{\sup_{t\in [0,T]} (X(t)+ g(t))> u\}$ as $u\to\infty$, 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.

28 pages