Multidimensional Brownian risk models with random trend

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 $a_1,\dots, a_d$.