## Electronic Journal of Probability

### The Vervaat transform of Brownian bridges and Brownian motion

#### 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 of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale property. The same study is done for the Vervaat transform of unconditioned Brownian motion, the expectation and variance of which are also derived.

#### Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 51, 31 pp.

Dates
Accepted: 6 May 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067157

Digital Object Identifier
doi:10.1214/EJP.v20-3744

Mathematical Reviews number (MathSciNet)
MR3347920

Zentralblatt MATH identifier
1327.60035

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Lupu, Titus; Pitman, Jim; Tang, Wenpin. The Vervaat transform of Brownian bridges and Brownian motion. Electron. J. Probab. 20 (2015), paper no. 51, 31 pp. doi:10.1214/EJP.v20-3744. https://projecteuclid.org/euclid.ejp/1465067157

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