Skorohod-reflection of Brownian Paths and BES^3

TL;DR

This paper provides an elementary proof that the difference of two Skorohod-reflected Brownian motions, driven by an independent Brownian motion, is a three-dimensional Bessel process, clarifying a previously established result.

Contribution

It offers a new, simpler proof of the known result that the difference of two reflected Brownian motions forms a BES^3 process.

Findings

The difference process Z(t) is a BES^3 process.

An elementary proof of the BES^3 characterization is provided.

The result confirms the structure of reflected Brownian motions in relation to Bessel processes.

Abstract

Let B(t), X(t) and Y(t) be independent standard 1d Borwnian motions. Define X^+(t) and Y^-(t) as the trajectories of the processes X(t) and Y(t) pushed upwards and, respectively, downwards by B(t), according to Skorohod-reflection. In a recent paper, Jon Warren proves inter alia that Z(t):= X^+(t)-Y^-(t) is a three-dimensional Bessel-process. In this note, we present an alternative, elementary proof of this fact.