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

Abstract

For $\{B_H(t)=(B_{H,1}(t),\dots ,B_{H,d}(t){)}^{\top },t\ge 0\}$, where $\{B_{H,i}(t),t\ge 0\},1\le i\le d$ 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 $\mu =({\mu }_1,\dots ,{\mu }_d{)}^{\top }\in R^d$, $\nu =({\nu }_1,\dots ,{\nu }_d{)}^{\top }\in R^d$ are such that there exists some $1\le i\le d$ such that ${\mu }_i>0,{\nu }_i>0.$

21 pages