Extremal behaviour of hitting a cone by correlated Brownian motion with drift

This paper derives an exact asymptotic expression for $${\mathbb{P}}_{{\mathbf{x}}_u}\{{\mathrm{\exists }}_{t\ge 0}\mathbf{X}(t)-\mathit{\mu }t\in \mathcal{U}\},\mathrm{a}\mathrm{s}u\to \mathrm{\infty },$$ where $\mathbf{X}(t)=(X_1(t),\dots ,X_d(t){)}^{\mathrm{\top }},t\ge 0$ is a correlated $d$-dimensional Brownian motion starting at the point ${\mathbf{x}}_u=-\mathit{\alpha }u$ with $\boldsymbol{\alpha}\in $\mathbb{R}$^d$,$\boldsymbol{\mu} \in $\mathbb{R}$^d$and$$\mathcal{U}$=\prod_{i=1}^d [0,\infty)$. The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to $\mathcal{U}$, which depends on the dimension-reduction phenomena.