Electronic Journal of Probability

The Vervaat transform of Brownian bridges and Brownian motion

Titus Lupu, Jim Pitman, and Wenpin Tang

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]

Brownian quartet Bessel bridges/processes Markov property path decomposition semi-martingale property size-biased sampling Vervaat transform

This work is licensed under aCreative Commons Attribution 3.0 License.


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

Export citation


  • Abramson, Josh; Pitman, Jim; Ross, Nathan; Uribe Bravo, Geronimo. Convex minorants of random walks and Lévy processes. Electron. Commun. Probab. 16 (2011), 423–434.
  • Aldous, David J. Brownian excursion conditioned on its local time. Electron. Comm. Probab. 3 (1998), 79–90 (electronic).
  • Aldous, David; Pitman, Jim. The standard additive coalescent. Ann. Probab. 26 (1998), no. 4, 1703–1726.
  • Sami Assaf, Noah Forman, and Jim Pitman. The quantile transform of a simple walk. 2013. ARXIV1307.4967.
  • Jacques. Azéma and Marc. Yor. Étude d'une martingale remarquable. In Séminaire de Probabilités, XXIII, volume 1372 of Lecture Notes in Math., pages 88–130. Springer, Berlin, 1989.
  • Beiglböck, Mathias; Schachermayer, Walter; Veliyev, Bezirgen. A direct proof of the Bichteler-Dellacherie theorem and connections to arbitrage. Ann. Probab. 39 (2011), no. 6, 2424–2440.
  • Beiglböck, M.; Siorpaes, P. Riemann-integration and a new proof of the Bichteler-Dellacherie theorem. Stochastic Process. Appl. 124 (2014), no. 3, 1226–1235.
  • Bertoin, Jean. A fragmentation process connected to Brownian motion. Probab. Theory Related Fields 117 (2000), no. 2, 289–301.
  • Bertoin, Jean; Chaumont, Loíc; Pitman, Jim. Path transformations of first passage bridges. Electron. Comm. Probab. 8 (2003), 155–166 (electronic).
  • Bertoin, J.; Chaumont, L.; Yor, M. Two chain-transformations and their applications to quantiles. J. Appl. Probab. 34 (1997), no. 4, 882–897.
  • Biane, Ph. Relations entre pont et excursion du mouvement brownien réel. (French) [Relations between Brownian bridge and excursion] Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 1, 1–7.
  • Biane, Ph.; Yor, M. Quelques précisions sur le méandre brownien. (French) [Some refinements of results on the Brownian meander] Bull. Sci. Math. (2) 112 (1988), no. 1, 101–109.
  • Bichteler, Klaus. Stochastic integration and $L^{p}$-theory of semimartingales. Ann. Probab. 9 (1981), no. 1, 49–89.
  • Billingsley, Patrick. Probability and measure. Third edition. Wiley Series in Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995. xiv+593 pp. ISBN: 0-471-00710-2
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
  • Chassaing, Philippe; Janson, Svante. A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0. Ann. Probab. 29 (2001), no. 4, 1755–1779.
  • Chaumont, L. A path transformation and its applications to fluctuation theory. J. London Math. Soc. (2) 59 (1999), no. 2, 729–741.
  • Chaumont, L. An extension of Vervaat's transformation and its consequences. J. Theoret. Probab. 13 (2000), no. 1, 259–277.
  • Loíc Chaumont and Gerónimo Uribe Bravo. Shifting processes with cyclically exchangeable increments at random. 2014. ARXIV1405.1335.
  • Chung, Kai Lai. Excursions in Brownian motion. Ark. Mat. 14 (1976), no. 2, 155–177.
  • Erhan Cinlar, Jean Jacod, Philip Protter, and Michael Sharpe. Semimartingales and Markov processes. Z. Wahrsch. Verw. Gebiete, 54(2):161–219, 1980.
  • Dassios, Angelos. The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options. Ann. Appl. Probab. 5 (1995), no. 2, 389–398.
  • Dassios, Angelos. Sample quantiles of stochastic processes with stationary and independent increments. Ann. Appl. Probab. 6 (1996), no. 3, 1041–1043.
  • Dassios, Angelos. On the quantiles of Brownian motion and their hitting times. Bernoulli 11 (2005), no. 1, 29–36.
  • Dellacherie, C. Un survol de la théorie de l'intégrale stochastique. (French) [A survey of the theory of the stochastic integral] Stochastic Process. Appl. 10 (1980), no. 2, 115–144.
  • Dellacherie, Claude; Meyer, Paul-André. Probabilities and potential. B. Theory of martingales. Translated from the French by J. P. Wilson. North-Holland Mathematics Studies, 72. North-Holland Publishing Co., Amsterdam, 1982. xvii+463 pp. ISBN: 0-444-86526-8
  • Denisov, I. V. Random walk and the Wiener process considered from a maximum point. (Russian) Teor. Veroyatnost. i Primenen. 28 (1983), no. 4, 785–788.
  • Embrechts, P.; Rogers, L. C. G.; Yor, M. A proof of Dassios' representation of the $\alpha$-quantile of Brownian motion with drift. Ann. Appl. Probab. 5 (1995), no. 3, 757–767.
  • Feller, William. An introduction to probability theory and its applications. Vol. I. Third edition John Wiley & Sons, Inc., New York-London-Sydney 1968 xviii+509 pp.
  • Patrick Fitzsimmons. Excursions above the minimum for diffusions. 1985. ARXIV1308.5189.
  • Fitzsimmons, Pat; Pitman, Jim; Yor, Marc. Markovian bridges: construction, Palm interpretation, and splicing. Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), 101–134, Progr. Probab., 33, Birkhäuser Boston, Boston, MA, 1993.
  • Forman, Noah Mills. Instruction sets for walks and the quantile path transformation. Thesis (Ph.D.)-University of California, Berkeley. ProQuest LLC, Ann Arbor, MI, 2013. 91 pp. ISBN: 978-1303-83567-4
  • Fourati, Sonia. Vervaat et Lévy. (French) [Vervaat and Levy] Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 3, 461–478.
  • Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. With one CD-ROM (Windows, Macintosh and UNIX). Seventh edition. Elsevier/Academic Press, Amsterdam, 2007. xlviii+1171 pp. ISBN: 978-0-12-373637-6; 0-12-373637-4
  • Iglehart, Donald L. Random walks with negative drift conditioned to stay positive. J. Appl. Probability 11 (1974), 742–751.
  • Imhof, J.-P. Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21 (1984), no. 3, 500–510.
  • Le Gall, Jean-François; Weill, Mathilde. Conditioned Brownian trees. Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 4, 455–489.
  • Titus Lupu. Poissonian ensembles of loops of one-dimensional diffusions. 2013. ARXIV1302.3773.
  • Miermont, Grégory. Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps. Electron. J. Probab. 6 (2001), no. 14, 33 pp. (electronic).
  • Perman, Mihael; Pitman, Jim; Yor, Marc. Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 (1992), no. 1, 21–39.
  • Pitman, Jim. Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times. Electron. J. Probab. 4 (1999), no. 11, 33 pp. (electronic).
  • Pitman, J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002. With a foreword by Jean Picard. Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
  • Pitman, Jim; Ross, Nathan. The greatest convex minorant of Brownian motion, meander, and bridge. Probab. Theory Related Fields 153 (2012), no. 3-4, 771–807.
  • Pitman, Jim; Yor, Marc. Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. Itô's stochastic calculus and probability theory, 293–310, Springer, Tokyo, 1996.
  • Port, Sidney C. An elementary probability approach to fluctuation theory. J. Math. Anal. Appl. 6 1963 109–151.
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4
  • Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1991.
  • Schweinsberg, Jason. Applications of the continuous-time ballot theorem to Brownian motion and related processes. Stochastic Process. Appl. 95 (2001), no. 1, 151–176.
  • Uribe Bravo, Geronimo. Bridges of Lévy processes conditioned to stay positive. Bernoulli 20 (2014), no. 1, 190–206.
  • Vervaat, Wim. A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 (1979), no. 1, 143–149.
  • Vigon, Vincent. (Homogeneous) Markovian bridges. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 3, 875–916.
  • Wendel, J. G. Order statistics of partial sums. Ann. Math. Statist. 31 1960 1034–1044.
  • Williams, David. Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 1970 871–873.
  • Williams, David. Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. (3) 28 (1974), 738–768.
  • Yen, Ju-Yi; Yor, Marc. Local times and excursion theory for Brownian motion. A tale of Wiener and Itô measures. Lecture Notes in Mathematics, 2088. Springer, Cham, 2013. x+135 pp. ISBN: 978-3-319-01269-8; 978-3-319-01270-4
  • Yor, Marc. The distribution of Brownian quantiles. J. Appl. Probab. 32 (1995), no. 2, 405–416.