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Windings of planar processes, exponential functionals and Asian options

  • Wissem Jedidi (a1) and Stavros Vakeroudis (a2)


Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.


Corresponding author

* Postal address: Department of Statistics & OR, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia. Email address:
** Postal address: Department of Mathematics, University of the Aegean, Vourliotis Building, Office Y5, 83200 Karlovasi, Samos, Greece. Email address:


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[1]Alili, L., Dufresne, D. and Yor, M. (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, Biblioteca de la Revista Matematica, Madrid, pp. 314.
[2]André, D. (1887). Solution directe du problème résolu par M. Bertrand. C. R. Acad. Sci. Paris 105, 436437.
[3]Bertoin, J. and Werner, W. (1996). Stable windings. Ann. Prob. 24, 12691279.
[4]Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Prob. Surveys 2, 191212.
[5]Biane, P. and Yor, M. (1987). Valeurs principales associées aux temps locaux Browniens. Bull. Sci. Math. 111, 23101.
[6]Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities (Lecture Notes Statist. 76). Springer, New York.
[7]Bondesson, L. (2015). A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables. J. Theoret. Prob. 28, 10631081.
[8]Bosch, P. and Simon, T. (2015). On the infinite divisibility of inverse beta distributions. Bernoulli 21, 25522568.
[9]Bougerol, P. (1983). Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré B 19, 369391.
[10]DeBlassie, R. D. (1987). Exit times from cones in ℝn of Brownian motion. Prob. Theory Relat. Fields 74, 129.
[11]DeBlassie, R. D. (1988). Remark on exit times from cones in ℝn of Brownian motion. Prob. Theory Relat. Fields 79, 9597.
[12]Doney, R. A. and Vakeroudis, S. (2013). Windings of planar stable processes. In Séminaire de Probabilités XLV (Lecture Notes Math. 2078), Springer, Cham, pp. 277300.
[13]Dufresne, D. (2000). Laguerre series for Asian and other options. Math. Finance 10, 407428.
[14]Gallardo, L. (2008). Mouvement Brownien et Calcul d'Itô. Hermann, Paris.
[15]Graversen, S. E. and Vuolle-Apiala, J. (1986). α-self-similar Markov processes. Prob. Theory Relat. Fields 71, 149158.
[16]Itô, K. and McKeanH. P., Jr. H. P., Jr. (1965). Diffusion Processes and Their Sample Paths. Springer, Berlin.
[17]Kyprianou, A. E. and Vakeroudis, S. M. (2018). Stable windings at the origin. To appear in Stoch. Process. Appl..
[18]Liao, M. and Wang, L. (2011). Isotropic self-similar Markov processes. Stoch. Process. Appl. 121, 20642071.
[19]Matsumoto, H. and Yor, M. (2005). Exponential functionals of Brownian motion. I. Probability laws at fixed time. Prob. Surveys 2, 312347.
[20]Matsumoto, H. and Yor, M. (2005). Exponential functionals of Brownian motion. II. Some related diffusion processes. Prob. Surveys 2, 348384.
[21]Messulam, P. and Yor, M. (1982). On D. Williams' 'pinching method' and some applications. J. London Math. Soc. 2 26, 348364.
[22]Pitman, J. and Yor, M. (1986). Asymptotic laws of planar Brownian motion. Ann. Prob. 14, 733779.
[23]Port, S. C. and Stone, C. J. (1978). Brownian Motion and Classical Potential Theory. Academic Press, New York.
[24]Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.
[25]Schilling, R. L., Song, R. and Vondraček, Z. (2012). Bernstein Functions: Theory and Applications, 2nd edn. De Gruyter, Berlin.
[26]Spitzer, F. (1958). Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87, 187197.
[27]Vakeroudis, S. (2011). Nombres de tours de certains processus stochastiques plans et applications à la rotation d'un polymère. Doctoral thesis. Université Pierre et Marie Curie.
[28]Vakeroudis, S. (2012). Bougerol's identity in law and extensions. Prob. Surveys 9, 411437.
[29]Vakeroudis, S. (2012). On hitting times of the winding processes of planar Brownian motion and of Ornstein–Uhlenbeck processes, via Bougerol's identity. Theory Prob. Appl. 56, 485507.
[30]Vakeroudis, S. and Yor, M. (2012). Some infinite divisibility properties of the reciprocal of planar Brownian motion exit time from a cone. Electron. Commun. Prob. 17, 23.
[31]Vakeroudis, S. and Yor, M. (2013). Integrability properties and limit theorems for the exit time from a cone of planar Brownian motion. Bernoulli 19, 20002009.
[32]Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Operat. Res. 5, 6785.
[33]Williams, D. (1974). A simple geometric proof of Spitzer's winding number formula for 2-dimensional Brownian motion. Unpublished manuscript.
[34]Yor, M. (1980). Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson. Z. Wahrscheinlichkeitsth. 53, 7195.
[35]Yor, M. (1992). Sur les lois des fonctionnelles exponentielles du mouvement Brownien, considérées en certains instants aléatoires. C. R. Acad. Sci. Paris I 314, 951956.
[36]Yor, M. (1993). From planar Brownian windings to Asian options. Insurance Math. Econom. 13, 2334.
[37]Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.
[38]Yor, M. and Geman, H. (2001). Bessel processes, Asian options, and perpetuities. In Exponential Functionals of Brownian Motion and Related Processes, Springer, Berlin, pp. 6392.


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