  Let \(\mathbf B(t)=(B_1(t), \dots,B_d(t))^\top\), \(t\in[0,T]\), \(d\geq 2\)
be a \(d\)-dimensional Brownian motion with independent components and let
\(\mathbf \eta=(\eta_1,\dots,\eta_d)^\top\) be a random vector independent of
\(\mathbf B\) such that
  \[
  \mathbb{P}{\mathbf K_{1}\leq\mathbf\eta\leq\vk K_{2}}
  =\mathbb{P}{K_{11}\leq\eta_1\leq K_{21},\dots,K_{1d}\leq\eta_d\leq K_{2d}}=1,
  \]
  where \(\mathbf K_1=(K_{11},\dots,K_{1d})^\top\) and \(\vk
K_2=(K_{21},\dots,K_{2d})^\top\) are fixed \(d\)-dimensional vectors.
  The goal of this paper is to derive asymptotics of
  \[
  \mathbb{P}{\exists_{t\in[0,T]}: X_1(t)>a_1u,\dots,X_d(t)>a_du}, \ \ \mathbf
X(t)=\left(X_1(t),\dots,X_d(t)\right)^\top
  =A\mathbf B(t)-\mathbf\eta t
  \]
  as \(u\to\infty\) under certain restrictions on the random vector
\(\mathbf\eta\) and constants \(a_1,\dots, a_d\).
