Abstract
This article contains both a point process and a sequential description of the greatest convex minorant of Brownian motion on a finite interval. We use these descriptions to provide new analysis of various features of the convex minorant such as the set of times where the Brownian motion meets its minorant. The equivalence of these descriptions is non-trivial, which leads to many interesting identities between quantities derived from our analysis. The sequential description can be viewed as a Markov chain for which we derive some fundamental properties.
References
Abramowitz, M., Stegun, I. (eds.).: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1992). Reprint of the 1972 edition
Abramson, J., Pitman, J.: Concave majorants of random walks and related Poisson processes. Combin. Probab. Comput. (2010, to appear). http://arxiv.org/abs/1011.3262
Balabdaoui F., Pitman J.: The distribution of the maximal difference between Brownian bridge and its concave majorant. Bernoulli 17(1), 466–483 (2011)
Bass, R.F.: Markov processes and convex minorants. In: Seminar on Probability, XVIII. Lecture Notes in Math., vol. 1059, pp. 29–41. Springer, Berlin (1984)
Bertoin J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)
Bertoin J.: The convex minorant of the Cauchy process. Electron. Commun. Probab. 5, 51–55 (2000)
Bertoin J., Chaumont L., Pitman J.: Path transformations of first passage bridges. Electron. Commun. Probab. 8, 155–166 (2003)
Blumenthal R.M.: Weak convergence to Brownian excursion. Ann. Probab. 11(3), 798–800 (1983)
Carolan, C., Dykstra, R.: Characterization of the least concave majorant of Brownian motion, conditional on a vertex point, with application to construction. Ann. Inst. Statist. Math. 55(3), 487–497 (2003). doi:10.1007/BF02517802. http://dx.doi.org/10.1007/BF02517802
Chaumont L., Uribe Bravo G.: Markovian bridges: weak continuity and pathwise constructions. Ann. Probab. 39(2), 609–647 (2011)
Çinlar E.: Sunset over Brownistan. Stoch. Process. Appl. 40(1), 45–53 (1992)
Denisov I.V.: Random walk and the Wiener process considered from a maximum point. Teor. Veroyatnost. i Primenen. 28(4), 785–788 (1983)
Dufresne, D.: On the stochastic equation \({\mathcal{L}(X)={\mathcal{L}}[B(X+C)]}\) and a property of gamma distributions. Bernoulli 2(3), 287–291 (1996). doi:10.2307/3318525. http://dx.doi.org/10.2307/3318525
Fitzsimmons, P., Pitman, J., Yor, M.: Markovian bridges: construction, palm interpretation, and splicing. In: Çinlar, E., Chung, K., Sharpe, M. (eds.) Seminar on Stochastic Processes, 1992, pp. 101–134. Birkhäuser, Boston (1993). http://stat.berkeley.edu/users/pitman/370.pdf
Fitzsimmons, P.J.: Another look at Williams’ decomposition theorem. In: Seminar on Stochastic Processes, 1985 (Gainesville, Fla., 1985). Progr. Probab. Statist., vol. 12, pp. 79–85. Birkhäuser Boston, Boston (1986)
Freedman D.: Brownian Motion and Diffusion. Holden-Day, San Francisco (1971)
Goldie C.M.: Records, permutations and greatest convex minorants. Math. Proc. Cambridge Philos. Soc. 106(1), 169–177 (1989)
Greenwood P., Pitman J.: Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Probab. 12, 893–902 (1980)
Groeneboom P.: The concave majorant of Brownian motion. Ann. Probab. 11(4), 1016–1027 (1983)
Imhof J.P.: Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21(3), 500–510 (1984)
Le Gall, J.F.: Une approche élémentaire des théorèmes de décomposition de Williams. In: Séminaire de Probabilités, XX, 1984/85. Lecture Notes in Math., vol. 1204, pp. 447–464. Springer, Berlin (1986)
Meyn, S.P., Tweedie, R.L.: Markov chains and stochastic stability. In: Communications and Control Engineering Series. Springer, London (1993)
Nagasawa, M.: Stochastic processes in quantum physics. In: Monographs in Mathematics, vol. 94. Birkhäuser Verlag, Basel (2000)
Pitman J.: One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Probab. 7, 511–526 (1975)
Pitman, J.: Remarks on the convex minorant of Brownian motion. In: Seminar on Stochastic Processes, 1982, pp. 219–227. Birkhäuser, Boston (1983)
Pitman, J.: Combinatorial stochastic processes. In: Lecture Notes in Mathematics, vol. 1875. Springer-Verlag, Berlin (2006). Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard
Pitman, J., Uribe Bravo, G.: The convex minorant of a Lévy process. Ann. Probab. (2010, to appear). http://arxiv.org/abs/1011.3069
Suidan T.M.: Convex minorants of random walks and Brownian motion. Teor. Veroyatnost. i Primenen. 46(3), 498–512 (2001)
Williams D.: Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. Lond. Math. Soc. 3(28), 738–768 (1974)
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Pitman’s research supported in part by N.S.F. Grant DMS-0806118.
Rights and permissions
About this article
Cite this article
Pitman, J., Ross, N. The greatest convex minorant of Brownian motion, meander, and bridge. Probab. Theory Relat. Fields 153, 771–807 (2012). https://doi.org/10.1007/s00440-011-0385-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-011-0385-0
Keywords
- Brownian motion
- Path decompositions
- Convex minorant
Mathematics Subject Classification (2000)
- 60J65
- 60J05
- 60E99