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

TL;DR

This paper provides an exact asymptotic expression for the probability that a correlated Brownian motion with drift hits a cone, revealing dimension-reduction phenomena and analyzing the distribution of the first passage time.

Contribution

It derives a precise asymptotic formula for hitting probabilities of a cone by correlated Brownian motion with drift, highlighting the role of quadratic optimization and dimension reduction.

Findings

Asymptotic probability depends on solving a constrained quadratic optimization problem.

Dimension reduction occurs in certain cases, simplifying the problem.

Distribution of the first passage time is characterized asymptotically.

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.