Probability Surveys

Bougerol’s identity in law and extensions

Stavros Vakeroudis

Full-text: Open access

Abstract

We present a list of equivalent expressions and extensions of Bougerol’s celebrated identity in law, obtained by several authors. We recall well-known results and the latest progress of the research associated with this celebrated identity in many directions, we give some new results and possible extensions and we try to point out open questions.

Article information

Source
Probab. Surveys, Volume 9 (2012), 411-437.

Dates
First available in Project Euclid: 8 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.ps/1352385533

Digital Object Identifier
doi:10.1214/12-PS195

Mathematical Reviews number (MathSciNet)
MR3007208

Zentralblatt MATH identifier
1278.60125

Subjects
Primary: 60J65: Brownian motion [See also 58J65] 60J60: Diffusion processes [See also 58J65] 60-02: Research exposition (monographs, survey articles) 60G07: General theory of processes
Secondary: 60G15: Gaussian processes 60J25: Continuous-time Markov processes on general state spaces 60G46: Martingales and classical analysis 60E10: Characteristic functions; other transforms 60J55: Local time and additive functionals 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination 44A10: Laplace transform

Keywords
Bougerol’s identity time-change hyperbolic Brownian motion subordination Gauss-Laplace transform planar Brownian motion Ornstein-Uhlenbeck processes two-dimensional Bougerol’s identity local time multi-dimensional Bougerol’s identity Bougerol’s diffusion peacock convex order Bougerol’s process

Citation

Vakeroudis, Stavros. Bougerol’s identity in law and extensions. Probab. Surveys 9 (2012), 411--437. doi:10.1214/12-PS195. https://projecteuclid.org/euclid.ps/1352385533


Export citation

References

  • [1] L. Alili, D. Dufresne and M. Yor (1997). Sur l’identité de Bougerol pour les fonctionnelles exponentielles du mouvement Brownien avec drift. In Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers; Biblioteca de la Revista Matematica, Ibero-Americana, ed. M. Yor, p. 3-14.
  • [2] L. Alili and J.C. Gruet (1997). An explanation of a generalised Bougerol’s identity in terms of Hyperbolic Brownian Motion. In Exponential Functionals and Principal Values related to Brownian Motion. A collection of research papers; Biblioteca de la Revista Matematica, Ibero-Americana, ed. M. Yor, p. 15-33.
  • [3] L. Alili, H. Matsumoto and T. Shiraishi (2001). On a triplet of exponential Brownian functionals. Sém. Prob. XXXV, Lect. Notes in Mathematics, 1755, Springer, Berlin Heidelberg New York, p. 396-415.
  • [4] D. André (1887). Solution directe du problème résolu par M. Bertrand. C. R. Acad. Sci. Paris, 105, p. 436-437.
  • [5] J. Bertoin, D. Dufresne and M. Yor (2012). Some two-dimensional extensions of Bougerol’s identity in law for the exponential functional of linear Brownian motion. Preprint. ArXiv: 1201.1495.
  • [6] J. Bertoin, D. Dufresne and M. Yor (2012). A relationship between Bougerol’s generalized identity in law and Jacobi processes. In Preparation.
  • [7] J. Bertoin and W. Werner (1994). Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process. Sém. Prob. XXVIII, Lect. Notes in Mathematics, 1583, Springer, Berlin Heidelberg New York, p. 138-152.
  • [8] J. Bertoin and M. Yor (2012). Retrieving information from subordination. To appear in Feitschrift volume for Professor Prokhorov.
  • [9] P. Biane and M. Yor (1987). Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math., 111, p. 23-101.
  • [10] Ph. Bougerol (1983). Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré, 19, p. 369-391.
  • [11] L. Chaumont and M. Yor (2012). Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning. Cambridge University Press, 2nd Edition.
  • [12] D. Dufresne (2000). Laguerre Series for Asian and Other Options. Mathematical Finance, Vol. 10, n. 4, p. 407-428.
  • [13] D. Dufresne and M. Yor (2011). A two dimensional extension of Bougerol’s identity in law for theexponential of Brownian motion. Working paper No. 222. Centre for Actuarial Studies, University of Melbourne.
  • [14] R. Durrett (1982). A new proof of Spitzer’s result on the winding of 2-dimensional Brownian motion. Ann. Prob., 10, p. 244-246.
  • [15] L. Gallardo (2008). Mouvement Brownien et calcul d’Itô. Hermann.
  • [16] F. Hirsch, C. Profeta, B. Roynette and M. Yor (2011). Peacocks and associated martingales, with explicit constructions. Bocconi and Springer Series, vol. 3, Springer.
  • [17] F. Hirsch and B. Roynette (2012). A new proof of Kellerer’s theorem. ESAIM: Probability and Statistics, 16, p. 48-60.
  • [18] K. Itô and H.P. McKean (1965). Diffusion Processes and their Sample Paths. Die Grundlehren der Mathematischen Wissenschaften, 125. Springer, Berlin Heidelberg New York.
  • [19] J. Jakubowski and M. Wisniewolski (2012). On hyperbolic Bessel processes and beyond. To appear in Bernoulli. Available at http://www.bernoulli-society.org/index.php/publications/bernoulli-journal/bernoulli-journal-papers
  • [20] H.G. Kellerer (1972). Markov-Komposition und eine Anwendung auf Martingale. Math. Ann., 198, p. 99-122.
  • [21] J. Lamperti (1972). Semi-stable Markov processes I. Z. Wahr. Verw. Gebiete, 22, p. 205-225.
  • [22] N.N. Lebedev (1972). Special Functions and their Applications. Revised edition, translated from the Russian and edited by Richard A. Silverman.
  • [23] P. Lévy (1980). Œuvres de Paul Lévy, Processus Stochastiques, Vol. IV. Paris: Gauthier-Villars. Published under the direction of D. Dugué with the collaboration of Paul Deheuvels and Michel Ibéro.
  • [24] H. Matsumoto and M. Yor (1998). On Bougerol and Dufresne’s identities for exponential Brownian functionals. Proc. Japan Acad. Ser. A Math. Sci., Volume 74, n. 10, p. 152-155.
  • [25] H. Matsumoto and M. Yor (2005). Exponential functionals of Brownian motion, I: Probability laws at fixed time. Probab. Surveys, Volume 2, p. 312-347.
  • [26] P. Messulam and M. Yor (1982). On D. Williams’ “pinching method” and some applications. J. London Math. Soc., 26, p. 348-364.
  • [27] D. Revuz and M. Yor (1999). Continuous Martingales and Brownian Motion. 3rd ed., Springer, Berlin.
  • [28] F. Spitzer (1958). Some theorems concerning two-dimensional Brownian Motion. Trans. Amer. Math. Soc. 87, p. 187-197.
  • [29] S. Vakeroudis (2011). Nombres de tours de certains processus stochastiques plans et applications à la rotation d’un polymère. (Windings of some planar Stochastic Processes and applications to the rotation of a polymer). PhD Dissertation, Université Pierre et Marie Curie (Paris VI), April 2011.
  • [30] S. Vakeroudis (2011). On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol’s identity. Teor. Veroyatnost. i Primenen., 56 (3), p. 566-591. Published also in SIAM Theory Probab. Appl. (2012) 56 (3), p. 485-507.
  • [31] S. Vakeroudis (2012). On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a Stable process. Preprint. ArXiv: 1209.4027.
  • [32] S. Vakeroudis and M. Yor (2012). Integrability properties and Limit Theorems for the first exit times from a cone of planar Brownian motion. To appear in Bernoulli. ArXiv: 1201.2716.
  • [33] S. Vakeroudis and M. Yor (2012). Some infinite divisibility properties of the reciprocal of planar Brownian motion exit time from a cone. Electron. Commun. Probab., 17, Paper No. 23.
  • [34] E.B. Vinberg (1993). Geometry II, Spaces of constant curvature. Encyclopædia of Math. Sciences, 29, Springer.
  • [35] J. Warren and M. Yor (1998). The brownian burglar: conditioning brownian motion by its local time process. Sém. Prob. XXXII, Lect. Notes in Mathematics, 1583, Springer, Berlin Heidelberg New York, p. 328-342.
  • [36] D. Williams (1974). A simple geometric proof of Spitzer’s winding number formula for 2-dimensional Brownian motion. University College, Swansea. Unpublished.
  • [37] M. Yor (1980). Loi de l’indice du lacet Brownien et Distribution de Hartman-Watson. Z. Wahrsch. verw. Gebiete, 53, p. 71-95.
  • [38] M. Yor (1992). On some Exponential Functionals of Brownian Motion. Adv. Appl. Prob., 24, n. 3, p. 509-531.
  • [39] M. Yor (1997). Generalized meanders as limits of weighted Bessel processes, and an elementary proof of Spitzer’s asymptotic result on Brownian windings. Studia Scient. Math. Hung. 33, p. 339-343.
  • [40] M. Yor (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer Finance. Springer-Verlag, Berlin.