Extremes of vector-valued Gaussian processes with Trend

Let X(t) (X-1(t),..., X-n (t)), t is an element of T subset of IR be a centered vector-valued Gaussian process with independent components and continuous trajectories, and h(t) = h(1)(t), ..., h(n)(t)), t is an element of T be a vector -valued continuous function. We investigate the asymptotics of P{sup(t is an element of T) min(1 <= i <= n) (X-i(t) + h(i)(t)) > u} as u -> infinity. 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. (C) 2018 Elsevier Inc. All rights reserved.