Extremes of vector-valued Gaussian processes with Trend

TL;DR

This paper derives the asymptotic behavior of the probability that the minimum of a vector-valued Gaussian process with trend exceeds a high threshold, with applications to ruin probabilities in risk models.

Contribution

It provides new asymptotic results for the supremum of vector Gaussian processes with trends, including specific cases with locally-stationary structures and varying variances.

Findings

Asymptotic formulas for the probability of the process exceeding high thresholds.

Exact asymptotics for simultaneous ruin probabilities.

Application to Gaussian risk models with explicit ruin time calculations.

Abstract

Let $X(t)=(X_1(t), \dots, X_n(t)), t\in \mathcal{T}\subset \mathbb{R} $ be a centered vector-valued Gaussian process with independent components and continuous trajectories, and $h(t)=(h_1(t),\dots, h_n(t)), t\in \mathcal{T} $ be a vector-valued continuous function. We investigate the asymptotics of $$\mathbb{P}\left(\sup_{t\in \mathcal{T} } \min_{1\leq i\leq n}(X_i(t)+h_i(t))>u\right)$$ as $u\to\infty$. 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.