Extremes of threshold-dependent Gaussian processes

Abstract

In this paper, we are concerned with the asymptotic behavior, as $$u\rightarrow\infty$$ , of $$\text{P}\left\{ {{{\sup }_{t \in [0,T]}}{X_u}(t) > u} \right\}$$ , 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 $$\text{P}\left\{ {{{\sup }_{t \in [0,T]}}(X(t) + g(t)) > u} \right\}$$ , as $$u\rightarrow\infty$$ , 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.