Extremes of vector-valued Gaussian processes with Trend

Let X(t)=(X₁(t),…,X_n(t)),t∈T⊂R be a centered vector-valued Gaussian process with independent components and continuous trajectories, and h(t)=(h₁(t),…,h_n(t)),t∈T be a vector-valued continuous function. We investigate the asymptotics of P{supt∈T⁡min1≤i≤n⁡(X_i(t)+h_i(t))>u} as u→∞. As an illustration to the derived results we analyze two important classes of X(t): with locally-stationary structure and with varying variances of the coordinates, and calculate exact asymptotics of simultaneous ruin probability and ruin time in a Gaussian risk model.