On Vervaat transform of Brownian bridges and Brownian motion

TL;DR

This paper investigates the Vervaat transform applied to Brownian motion and bridges, analyzing its distribution, path decompositions, and semimartingale properties, especially when endpoints differ, extending understanding of quantile transforms in stochastic processes.

Contribution

It provides a detailed analysis of the Vervaat transform for Brownian motion and bridges with arbitrary endpoints, including distribution descriptions and semimartingale properties, which was not previously known.

Findings

Distribution described via path decompositions

Semimartingale properties characterized

Expectation and variance derived

Abstract

For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(\tau(f)+t \mod1)+f(1)1_{\{t+\tau(f) \geq 1\}}-f(\tau(f))$, where $\tau(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms for random walks and Brownian motion, we study the Vervaat transform of Brownian motion and Brownian bridges with arbitary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decompositions and study its semimartingale properties. The expectation and variance of the Vervaat transform of Brownian motion are also derived.