Simultaneous ruin probability for multivariate gaussian risk model

Abstract

Let $\text{Z}(t)=(Z_1(t),\dots ,Z_d(t){)}^{\top },t\in \mathbb{R}$ 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 $\text{X}(t)=A\text{Z}(t),t\in \mathbb{R}$, where $A$ is a nonsingular $d\times d$ real-valued matrix, $\text{u},\text{c}\in {\mathbb{R}}^d$ 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 $(u_1,...,u_d{)}^{\top }=(u{a}_1,...,u{a}_d{)}^{\top }$ and $u\to \infty$.