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.
Categories: math.PR