Some points on Vervaat's transform of Brownian bridges and Brownian motion

TL;DR

This paper provides an alternative proof for the path decomposition of Vervaat's transform of Brownian bridges and motion, and resolves open questions about their semi-martingale properties, advancing understanding of these stochastic processes.

Contribution

It offers a new proof for the path decomposition and answers open questions about the semi-martingale decomposition of Vervaat's transform.

Findings

Confirmed the path decomposition properties of Vervaat's transform.

Resolved open questions regarding semi-martingale decomposition.

Enhanced theoretical understanding of Brownian bridges and motion transformations.

Abstract

In a recent work J. Pitman and W. Tang defined the Vervaat's transform for a Brownian bridge with two different endpoints and for a Brownian motion between times $0$ and $1$. They proved some path decomposition properties for these Vervaat's transforms and raised open questions on their semi-martingale decomposition. In this paper we give an alternative proof for the path decomposition and answer the open questions.