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2015 The Vervaat transform of Brownian bridges and Brownian motion
Titus Lupu, Jim Pitman, Wenpin Tang
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Electron. J. Probab. 20: 1-31 (2015). DOI: 10.1214/EJP.v20-3744


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.


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Titus Lupu. Jim Pitman. Wenpin Tang. "The Vervaat transform of Brownian bridges and Brownian motion." Electron. J. Probab. 20 1 - 31, 2015.


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

zbMATH: 1327.60035
MathSciNet: MR3347920
Digital Object Identifier: 10.1214/EJP.v20-3744

Primary: 60C05
Secondary: 60J60, 60J65


Vol.20 • 2015
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