Hitting times to spheres of Brownian motions with and without drifts

TL;DR

This paper derives explicit formulas for the densities and Laplace transforms of the first hitting times to spheres for Brownian motions with and without drifts, providing new insights into their joint distributions.

Contribution

It introduces novel explicit expressions for the joint distributions and Laplace transforms of hitting times and positions, differing from existing literature.

Findings

Explicit formulas for hitting time densities with drifts

Laplace transforms of joint distributions derived

New expressions differ from previous known results

Abstract

Explicit formulae for the densities of the first hitting times to the sphere of Brownian motions with drifts are given. We need to consider the joint distributions of the first hitting times to the sphere and the hitting positions of the standard Brownian motion and explicit expression for their Laplace transforms are given, which are different from the known formulae in the literature and are of independnt interest.