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On the excision of Brownian bridge paths

Gabriel Berzunza Ojeda, Ju-Yi Yen

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.

On the excision of Brownian bridge paths

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 -dimensional Bessel process . This paper is specifically motivated by Pitman and Yor (2003), who showed that a 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 -dimensional Bessel bridge.
Paper Structure (18 sections, 21 theorems, 96 equations, 1 figure)

This paper contains 18 sections, 21 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some known Brownian path transformations
  3. Deterministic path transformations
  4. From bridges to time-reversed meander-type functions
  5. Excising excursions from bridges
  6. Excising excursions from time-reversed meander-type functions
  7. Excising excursions from continuous functions
  8. Excising Excursions from the standard Brownian motion
  9. The law of $\tau_{x}$
  10. The law of $X_{x}$
  11. Proof of the main result
  12. \ref{['Step1']} From the bridge to the meander.
  13. \ref{['Step2']} Weighted expectation via conditioning.
  14. \ref{['Step3']} First-passage bridge and identification of the kernel.
  15. Proof of Proposition \ref{['Pro3']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

For all positive or bounded measurable function $G: \mathbf{C}([0,1], \mathbb{R}) \rightarrow \mathbb{R}$, we have that

Figures (1)

  • Figure 1: Excising excursions of a Brownian bridge below $M^{\mathrm{br}}(t)$. Excursions that reach level $0$ (red shading) are excised, while those that remain above $0$ (blue shading) are kept and concatenated.

Theorems & Definitions (52)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Lemma 2.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • Lemma 3.4
  • Definition 3.5: Regularity
  • ...and 42 more