Brownian Hitting to Spheres

TL;DR

This paper derives explicit formulas for the joint distribution of the hitting time and position of Brownian motion with drift to a sphere, and analyzes their asymptotic behavior using spherical harmonics.

Contribution

It provides new explicit formulas for the joint distribution of hitting time and position for Brownian motion with drift hitting a sphere, using spherical harmonics.

Findings

Explicit formulas for the joint distribution density.

Asymptotic behavior of the distribution function analyzed.

Results applicable to Brownian motion with constant drift.

Abstract

Let $S^{d-1}_r$ be the sphere in $\bR^d$ whose center is the origin and the radius is $r$, and $\sigma_r$ be the first hitting time to it of the standard Brownian motion $\{B_t\}_{t\geqq0}$, possibly with constant drift. The aim of this article is to show explicit formulae by means of spherical harmonics for the density of the joint distribution of $(\sigma_r,B_{\sigma_r})$ and to study the asymptotic behavior of the distribution function.