Simultaneous ruin probability for multivariate gaussian risk model

TL;DR

This paper derives tight bounds and exact asymptotics for the probability that a multivariate Gaussian risk process simultaneously exceeds certain thresholds over a time interval, extending risk analysis in multivariate Gaussian models.

Contribution

It provides new tight bounds and asymptotic formulas for the simultaneous ruin probability in multivariate Gaussian risk models with stationary increments.

Findings

Derived tight bounds for the ruin probability.

Obtained exact asymptotics as thresholds grow large.

Extended risk analysis to multivariate Gaussian processes.

Abstract

Let $\textbf{Z}(t)=(Z_1(t) ,\ldots, Z_d(t))^\top , t \in \mathbb{R}$ where $Z_i(t), t\in \mathbb{R}$, $i=1,...,d$ are mutually independent centered Gaussian processes with continuous sample paths a.s. and stationary increments. For $\textbf{X}(t)= A \textbf{Z}(t),\ t\in\mathbb{R}$, where $A$ is a nonsingular $d\times d$ real-valued matrix, $\textbf{u}, \textbf{c}\in\mathbb{R}^d$ and $T>0$ we derive tight bounds for \[ \mathbb{P}\left\{\exists_{t\in [0,T]}: \cap_{i=1}^d \{ X_i(t)- c_i t > u_i\}\right\} \] and find exact asymptotics as $(u_1,...,u_d)^{\top}= (u a_1,..., ua_d)^\top$ and $u\to\infty$.