  For $\{B_H(t)= (B_{H,1}(t), \ldots, B_{H,d}(t))^\top,t\ge0\}$, where
$\{B_{H,i}(t),t\ge 0\}, 1\le i\le d$ are mutually independent fractional
Brownian motions, we obtain the exact asymptotics of $$ \mathbb P (\exists t\ge
0: A B_{H}(t) - \mu t >\nu u), \ \ \ \ u\to\infty, $$ where $A$ is a
non-singular $d\times d$ matrix and $\mu=(\mu_1,\ldots, \mu_d)^\top\in R^d$,
$\nu=(\nu_1, \ldots, \nu_d)^\top \in R^d$ are such that there exists some $1\le
i\le d$ such that $\mu_i>0, \nu_i>0.$
