  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 $u\to\infty$.
