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On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Hiroyuki Matsumoto  and
Marc Yor
Hiroyuki Matsumoto*
Affiliation:
Nagoya University
Marc Yor*
Affiliation:
Université Pierre et Marie Curie, Paris
*
∗ Postal address: Graduate School of Human Informatics, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan. Email address: [email protected]
∗∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu, Case Courrier 188, F-75252 Paris Cedex 05, France.
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Abstract

Denote by αt(μ) the probability law of At(μ) = ∫0texp(2(Bs+μ s))ds for a Brownian motion {Bs, s ≥ 0}. It is well known that αt(μ) is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for αt(0) and αt(1), as well as an equally remarkable relationship between αt(μ) and αt(ν) for two different drifts μ and ν. In this paper, hinging on previous results about αt(μ), we give different proofs of Dufresne's results and present extensions of them for the processes {At(μ), t ≥ 0}.

Keywords

Brownian motions with driftsBrownian exponential functionalsdensities in closed form

MSC classification

Primary: 60J65: Brownian motion 60J25: Continuous-time Markov processes on general state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 184 - 206
Copyright
Copyright © Applied Probability Trust 2003 

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References

Alili, L. and Gruet, J.-C. (1997). An explanation of a generalized Bougerol's identity in terms of hyperbolic Brownian motion. In [ pp. 15–33] madrid. In Exponential Functionals and Principal Values Related to Brownian Motion, ed. Yor, M., Revista Matemática Iberoamericana, Madrid, pp. 15–33.Google Scholar
Alili, L., Dufresne, D. and Yor, M. (1997). Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement brownien avec drift. In [pp. 3–14]madrid. In Exponential Functionals and Principal Values Related to Brownian Motion, ed. Yor, M., Revista Matemática Iberoamericana, Madrid, pp. 3–14.Google Scholar
Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Statist. 5, 151–157.Google Scholar
Barrieu, P. (2002). Etude de la densité de Hartman–Watson. , Université Pierre et Marie Curie, Paris.Google Scholar
Baudoin, F. (2002). Further exponential generalization of Pitman's 2M-X theorem. Electron. Commun. Prob. 7, 37–46.Google Scholar
Bhattacharya, R., Thomann, E. and Waymire, E. (2001). A note on the distribution of integrals of geometric Brownian motion. Statist. Prob. Lett. 55, 187–192.Google Scholar
Bougerol, Ph. (1983). Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré Sec. B 19, 369–391.Google Scholar
Comtet, A., Monthus, C. and Yor, M. (1998). Exponential functionals of Brownian motion and disordered systems, J. Appl. Prob. 35, 255–271.Google Scholar
Donati-Martin, C., Matsumoto, H. and Yor, M. (2002). Some absolute continuity relationships for certain anticipative transformations of geometric Brownian motions. Publ. RIMS Kyoto Univ. 37, 295–326.CrossRefGoogle Scholar
Donati-Martin, C., Matsumoto, H. and Yor, M. (2002). The law of geometric Brownian motion and its integral, revisited; application to conditional moments. In Mathematical Finance (Bachelier Congress 2000), eds Geman, H., Madan, D., Pliska, S. R. and Vorst, T., Springer, Berlin, pp. 221–243.Google Scholar
Dufresne, D. (1996). On the stochastic equation L(X)= L[B(X+C)] and a property of gamma distributions. Bernoulli 2, 287–291.Google Scholar
Dufresne, D. (1998). Algebraic properties of beta and gamma distributions, and applications. Adv. Appl. Math. 20, 285–299.CrossRefGoogle Scholar
Dufresne, D. (2001). An affine property of the reciprocal Asian option process. Osaka J. Math. 38, 379–381.Google Scholar
Dufresne, D. (2001). The integral of geometric Brownian motion. Adv. Appl. Prob. 33, 223–241.CrossRefGoogle Scholar
Geman, H. and Yor, M. (1993). Bessel processes, Asian options, and perpetuities. Math. Finance 3, 349–375.Google Scholar
Gruet, J.-C. (1996). Semi-groupe du mouvement brownien hyperbolique. Stoch. Stoch. Reports 56, 53–61.Google Scholar
Hariya, Y. and Yor, M. (2002). Limiting distributions associated with moments of exponential Brownian functionals. Preprint, RIMS, University of Kyoto, and Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, Paris.Google Scholar
Ikeda, N. and Matsumoto, H. (1999). Brownian motion on the hyperbolic plane and Selberg trace formula. J. Funct. Anal. 162, 63–110.Google Scholar
Kawazu, K. and Tanaka, H. (1991). On the maximum of a diffusion process in a drifted Brownian environment. In Séminaire de Probabilités XXVII (Lecture Notes Math. 1557), Springer, Berlin, pp. 78–85.Google Scholar
Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.Google Scholar
Letac, G. and Wesolowski, J. (2000). An independence property for the product of GIG and gamma laws. Ann. Prob. 28, 1371–1383.Google Scholar
Linetsky, V. (2001). A closed-form formula for the arithmetic Asian option. Preprint.Google Scholar
Matsumoto, H. and Yor, M. (1999). A version of Pitman's 2M-X theorem for geometric Brownian motions. C. R. Acad. Sci. Paris Sér. I Math. 328, 1067–1074.Google Scholar
Matsumoto, H. and Yor, M. (2000). An analogue of Pitman's 2M-X theorem for exponential Wiener functionals, Part I: A time inversion approach. Nagoya Math. J. 159, 125–166.Google Scholar
Matsumoto, H. and Yor, M. (2001). A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration. Osaka J. Math. 38, 383–398.Google Scholar
Matsumoto, H. and Yor, M. (2001). An analogue of Pitman's 2M-X theorem for exponential Wiener functionals, Part II: The role of the generalized inverse Gaussian laws. Nagoya Math. J. 162, 65–86.Google Scholar
Matsumoto, H. and Yor, M. (2002). Interpretation via Brownian motion of some independence properties between GIG and gamma variables. To appear in Statist. Prob. Lett.Google Scholar
Schröder, M., (1999). On the valuation of arithmetic-average Asian options: explicit formulas. Preprint, Universität Mannheim.Google Scholar
Schröder, M., (2002). Mathematical ramifications of option valuation: the case of the Asian option. Habilitationsschrift, Universität Mannheim.Google Scholar
Stieltjes, T. J. (1894). Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, J1–J122 and 9, A5–A47. Reprinted in: Ann. Fac. Sci. Toulouse (6) 4 (1995), J1–J122, A5–A47.CrossRefGoogle Scholar
Vallois, P. (1991). La loi gaussienne inverse généralisée comme premier ou dernier temps de passage de diffusions. Bull. Sci. Math. 115, 301–368.Google Scholar
Yano, K. (2001). A generalization of the Buckdahn–Föllmer formula for composite transformations defined by finite dimensional substitution. Preprint. To appear in J. Math. Kyoto Univ. Google Scholar
Yor, M. (1980). Loi de l'indice du lacet brownien, et distribution de Hartman–Watson. Z. Wahrscheinlichkeitsth. 53, 71–95.CrossRefGoogle Scholar
Yor, M. (1992). On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509–531.CrossRefGoogle Scholar
Yor, M. (1992). Sur les lois des fonctionnelles exponentielles du mouvement brownien, considérées en certains instants aléatoires. C. R. Acad. Sci. Paris Sér. I. Math. 314, 951–956.Google Scholar
Yor, M. (ed.) (1997). Exponential Functionals and Principal Values Related to Brownian Motion. Revista Matemática Iberoamericana, Madrid.Google Scholar
Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.CrossRefGoogle Scholar