On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes

TL;DR

This paper derives the distribution of hitting times for affine boundaries by reflecting Brownian motion and Bessel processes, utilizing Doob's formula, time inversion, and time reversal properties, with applications to Bessel bridges.

Contribution

It introduces a novel application of Doob's formula and time properties to compute hitting time distributions for affine boundaries by reflecting Brownian motion and Bessel processes.

Findings

Distribution function for hitting times of affine boundaries by reflecting Brownian motion

Application of methodology to three-dimensional Bessel processes

Equivalence of crossing probabilities and staying below fixed values for Bessel bridges

Abstract

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary $t\mapsto a+bt,\ a\geq 0,\,b\in \R,$ by a reflecting Brownian motion. The main tool hereby is Doob's formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter $\delta>0$ and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.