Probability of entering an orthant by correlated fractional Brownian motion with drift: exact asymptotics

For \(\{\varvec{B}_{H}(t)= (B_{H,1}(t) ,\ldots ,B_{H,d}(t))^{{\top }},t\ge 0\}\), 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

$$\mathbb P (\exists t\ge 0: A \varvec{B}_{H}(t) - \varvec{\mu }t >\varvec{\nu }u), \ \ \ \ u\rightarrow \infty ,$$

where A is a non-singular \(d\times d\) matrix and \(\varvec{\mu }=(\mu _1,\ldots , \mu _d)^{{\top }}\in \mathbb {R}^d\), \(\varvec{\nu }=(\nu _1, \ldots , \nu _d)^{{\top }} \in \mathbb {R}^d\) are such that there exists some \(1\le i\le d\) such that \(\mu _i>0, \nu _i>0.\)