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 \(\mathbf η=(η_1,\dots,η_d)^\top\) be a random vector independent of \(\mathbf B\) such that \[ \mathbb{P}{\mathbf K_{1}\leq\mathbfη\leq\vk K_{2}} =\mathbb{P}{K_{11}\leqη_1\leq K_{21},\dots,K_{1d}\leqη_d\leq K_{2d}}=1, \] 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 \[ \mathbb{P}{\exists_{t\in[0,T]}: X_1(t)>a_1u,\dots,X_d(t)>a_du}, \ \ \mathbf X(t)=\left(X_1(t),\dots,X_d(t)\right)^\top =A\mathbf B(t)-\mathbfηt \] as \(u\to\infty\) under certain restrictions on the random vector \(\mathbfη\) and constants \(a_1,\dots, a_d\).