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

Abstract: This paper derives an exact asymptotic expression for [ \mathbb{P}{\mathbf{x}_u}{\exists{t\ge0} \mathbf{X}(t)- \boldsymbol{\mu}t\in \mathcal{U} }, \ \ {\rm as}\ \ u\to\infty, ] where $\mathbf{X}(t)=(X_1(t),\ldots,X_d(t))^\top,t\ge0$ is a correlated $d$-dimensional Brownian motion starting at the point $\mathbf{x}u=-\boldsymbol{\alpha}u$ 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.