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Bougerol’s identity in law and extensions. (English) Zbl 1278.60125
Summary: 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.

MSC:
60J65 Brownian motion
60J60 Diffusion processes
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60G07 General theory of stochastic processes
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
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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. · Zbl 0905.60059
[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. · Zbl 0911.60069
[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. · Zbl 0981.60080 · numdam:SPS_2001__35__396_0 · eudml:114075
[4] D. André (1887). Solution directe du problème résolu par M. Bertrand. C. R. Acad. Sci. Paris , 105 , p. 436-437. · JFM 19.0200.05
[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. · Zbl 1303.60073
[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. · Zbl 0814.60080 · numdam:SPS_1994__28__138_0 · eudml:113869
[8] J. Bertoin and M. Yor (2012). Retrieving information from subordination. To appear in Feitschrift volume for Professor Prokhorov. · Zbl 1284.60084
[9] P. Biane and M. Yor (1987). Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. , 111 , p. 23-101. · Zbl 0619.60072
[10] Ph. Bougerol (1983). Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré , 19 , p. 369-391. · Zbl 0533.60010 · numdam:AIHPB_1983__19_4_369_0 · eudml:77219
[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. · Zbl 1246.60001
[12] D. Dufresne (2000). Laguerre Series for Asian and Other Options. Mathematical Finance , Vol. 10 , n. 4, p. 407-428. · Zbl 1014.91040 · doi:10.1111/1467-9965.00101
[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. · Zbl 0479.60081 · doi:10.1214/aop/1176993928
[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. · Zbl 1227.60001 · doi:10.1007/978-88-470-1908-9
[17] F. Hirsch and B. Roynette (2012). A new proof of Kellerer’s theorem. ESAIM: Probability and Statistics , 16 , p. 48-60. · Zbl 1277.60041 · doi:10.1051/ps/2011164
[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. · Zbl 0127.09503
[19] J. Jakubowski and M. Wisniewolski (2012). On hyperbolic Bessel processes and beyond. To appear in Bernoulli . Available at · www.bernoulli-society.org
[20] H.G. Kellerer (1972). Markov-Komposition und eine Anwendung auf Martingale. Math. Ann. , 198 , p. 99-122. · Zbl 0229.60049 · doi:10.1007/BF01432281 · eudml:162296
[21] J. Lamperti (1972). Semi-stable Markov processes I. Z. Wahr. Verw. Gebiete , 22 , p. 205-225. · Zbl 0274.60052 · doi:10.1007/BF00536091
[22] N.N. Lebedev (1972). Special Functions and their Applications. Revised edition, translated from the Russian and edited by Richard A. Silverman. · Zbl 0271.33001
[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. · Zbl 0942.60063 · doi:10.3792/pjaa.74.152
[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. · Zbl 1189.60150 · doi:10.1214/154957805100000159 · eudml:227045
[26] P. Messulam and M. Yor (1982). On D. Williams’ “pinching method” and some applications. J. London Math. Soc. , 26 , p. 348-364. · Zbl 0518.60088 · doi:10.1112/jlms/s2-26.2.348
[27] D. Revuz and M. Yor (1999). Continuous Martingales and Brownian Motion. 3rd ed., Springer, Berlin. · Zbl 0917.60006
[28] F. Spitzer (1958). Some theorems concerning two-dimensional Brownian Motion. Trans. Amer. Math. Soc. 87 , p. 187-197. · Zbl 0089.13601 · doi:10.2307/1993096
[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. · Zbl 1273.60103
[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 . · Zbl 1259.60097
[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. · Zbl 1259.60097
[34] E.B. Vinberg (1993). Geometry II, Spaces of constant curvature. Encyclopædia of Math. Sciences , 29 , Springer. · Zbl 0786.00008
[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. · Zbl 0924.60072 · numdam:SPS_1998__32__328_0 · eudml:113994
[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. · Zbl 0436.60057 · doi:10.1007/BF00531612
[38] M. Yor (1992). On some Exponential Functionals of Brownian Motion. Adv. Appl. Prob. , 24 , n. 3, p. 509-531. · Zbl 0765.60084 · doi:10.2307/1427477
[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. · Zbl 0909.60070
[40] M. Yor (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer Finance. Springer-Verlag, Berlin. · Zbl 0999.60004 · doi:10.1007/978-3-642-56634-9
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