Title: 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.