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

For \(\{\varvec{B}_{H}(t)= (B_{H,1}(t) ,\ldots ,B_{H,d}(t))^{{\top }},t\ge 0\}\) $\{{\mathit{B}}_H(t)={(B_{H,1}(t),\dots ,B_{H,d}(t))}^{\top },t\ge 0\}$ , where \(\{B_{H,i}(t),t\ge 0\}, 1\le i\le d\) $\{B_{H,i}(t),t\ge 0\},1\le i\le d$ are mutually independent fractional Brownian motions, we obtain the exact asymptotics of \(\mathbb P (\exists t\ge 0: A \varvec{B}_{H}(t) - \varvec{\mu }t >\varvec{\nu }u), \ \ \ \ u\rightarrow \infty ,\) $$\mathbb{P}(\exists t\ge 0:A{\mathit{B}}_H(t)-\mathit{\mu }t>\mathit{\nu }u),u\to \infty ,$$ where A is a non-singular \(d\times d\) $d\times d$ matrix and \(\varvec{\mu }=(\mu _1,\ldots , \mu _d)^{{\top }}\in \mathbb {R}^d\) $\mathit{\mu }={({\mu }_1,\dots ,{\mu }_d)}^{\top }\in {\mathbb{R}}^d$ , \(\varvec{\nu }=(\nu _1, \ldots , \nu _d)^{{\top }} \in \mathbb {R}^d\) $\mathit{\nu }={({\nu }_1,\dots ,{\nu }_d)}^{\top }\in {\mathbb{R}}^d$ are such that there exists some \(1\le i\le d\) $1\le i\le d$ such that \(\mu _i>0, \nu _i>0.\) ${\mu }_i>0,{\nu }_i>0.$