On the excision of Brownian bridge paths

TL;DR

This paper explores how excising certain excursions from a Brownian bridge relates to a 3-dimensional Bessel bridge, extending known path transformation techniques in stochastic processes.

Contribution

It demonstrates a novel excision procedure on Brownian bridges that connects to Bessel bridges, expanding the understanding of path transformations.

Findings

Excision of Brownian bridge excursions relates to Bessel bridge construction.

Establishes a connection between Brownian bridges and Bessel bridges via path excision.

Extends Pitman and Yor's excursion excision method to Brownian bridges.

Abstract

Path transformations are fundamental to the study of Brownian motion and related stochastic processes, offering elegant constructions of the Brownian bridge, meander, and excursion. Central to this theory is the well-established link between Brownian motion and the $3$-dimensional Bessel process ${\rm BES}(3)$. This paper is specifically motivated by Pitman and Yor (2003), who showed that a ${\rm BES}(3)$ process can be constructed by excising the excursions of a Brownian path below its past maximum that reach zero and concatenating the remaining excursions. Our main result shows that a similar excision procedure, when applied to a Brownian bridge, can be related to a $3$-dimensional Bessel bridge.