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

Abstract

For $\{B_H\left(t\right)={(B_{H,1}\left(t\right),\dots ,B_{H,d}\left(t\right))}^{\top },t\ge 0\}$ , where $\{B_{H,i}\left(t\right),t\ge 0\},1\le i\le d$ are mutually independent fractional Brownian motions, we obtain the exact asymptotics of $P(\exists t\ge 0:A{B}_H\left(t\right)-\mu t>\nu u),u\to \infty ,$ 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 ${\mu }_i>0,{\nu }_i>0.$

Keywords: Dimension reduction; Exact asymptotics; Extremes; Large deviations; Multi-dimensional fractional Brownian motion; Quadratic programming problem.