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

Abstract

This paper derives an exact asymptotic expression for \[ \mathbb{P}_{\mathbf{x}_u}\{\exists_{t\ge0} \mathbf{X}(t)- \boldsymbolμt\in \mathcal{U} \}, \ \ {\rm as}\ \ u\to\infty, \] where $\mathbf{X}(t)=(X_1(t),\dots ,X_d(t){)}^{\top },t\ge 0$ is a correlated $d$-dimensional Brownian motion starting at the point ${\mathbf{x}}_u=-\mathbf{\alpha }u$ with $\mathbf{\alpha }\in {\mathbb{R}}^d$, $\mathbf{\mu }\in {\mathbb{R}}^d$ 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.

32 pages

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