The Law of Geometric Brownian Motion and its Integral, Revisited; Application to Conditional Moments

  • Catherine Donati-Martin
  • Hiroyuki Matsumoto
  • Marc Yor
Part of the Springer Finance book series (FINANCE)


Part A of this paper is a transcript of the third author’s lecture at the Bachelier Conference, July 1st, 2000; it is a summary of our joint works on this topic.


Recurrence Formula Geometric Brownian Motion Bernoulli Number Conditional Moment Mathematical Finance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Alili, D. Dufresne et, M. Yor, 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, Ed. by M. Yor, Biblioteca de la Revista Matemàtica Iberoamericana, 1997.Google Scholar
  2. 2.
    L. Chaumont, D.G. Hobson and M. Yor, Some consequences of the cyclic exchangeability property for exponential functionals of Lévy processes, Séni. Prob. XXXV, Lec. Notes Math. 1755, 334–347, Springer-Verlag, 2001.Google Scholar
  3. 3.
    P. Carmona, F. Petit, and M. Yor, Exponential functionals of Lévy processes, in Lévy Processes: Theory and Applications, ed. by O.E. Barndorff-Nielsen, T. Mikosch and S.I. Resnick, 41–56, Birkhâuser, 2001.Google Scholar
  4. 4.
    C. Donati-Martin, H. Matsumoto and M. Yor, On striking identities about, the exponential functionals of the Brownian bridge and Brownian motion, Periodica Math. Hung., 41 (2000), 103–119.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    C. Donati-Martin, H. Matsumoto and M. Yor, On positive and negative moments of the integral of geometric Brownian motions, Stat,. Prob. Lett,., 49 (2000), 4. 5–52.MathSciNetGoogle Scholar
  6. 6.
    D. Dufresne, The distribution of perpetuity, with applications to risk theory and pension funding, Scand. Act,. J., 1990, 39–79.Google Scholar
  7. 7.
    D. Dufresne, An affine property of the reciprocal Asian option process, Osaka J. Math. 38 (2001), 379–381.Google Scholar
  8. 8.
    D. Dufresne, Laguerre series for Asian and other options, Math. Finance, 10 (2000), 407–428.Google Scholar
  9. 9.
    H. Geman and M. Yor, Bessel processes, Asian options, and perpetuities. Math. Finance, 3 (1993), 349–375.Google Scholar
  10. 10.
    N. O’Connell and M. Yor, Brownian analogues of Burke’s theorem, to appear in Stoch. Proc. Appl., 2001.Google Scholar
  11. 11.
    K. It,ô and H.P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer-Ver lag, Berlin, 1965.Google Scholar
  12. 12.
    D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall, London, 1996.Google Scholar
  13. 13.
    J. Lamperti, Semi-stable Markov processes I, Z.W., 22 (1972), 205–255.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    N.N. Lebedev, Special Functions and their Applications, Dover, New York, 1972.MATHGoogle Scholar
  15. 15.
    E.H. Lehmann, Testing Statistical Hypotheses, 2nd Ed., Wiley, New York, 1986.CrossRefMATHGoogle Scholar
  16. 16.
    H. Matsumoto and M. Yor, On Bougerol and Dufresne’s identities for exponential Brownian functionals, Proc. Japan Acad., 74 Ser.A (1998), 152–155.Google Scholar
  17. 17.
    H. Matsumoto and M. Yor, A version of Pitman’s 2M — X theorem for geometric Brownian motions, C. R. Acad. Sc. Paris, 328 (1999), 1067–1074.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    H. Matsumoto and M. Yor, A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration, Osaka J. Math. 38 (2001), 383–398.MathSciNetMATHGoogle Scholar
  19. 19.
    E.J. Pauwels and L.C.G. Rogers, Skew-product, decompositions of Brownian motions, Geometry of random motion, 237–262, Contemp. Math., 73, AMS, Providence, 1988.Google Scholar
  20. 20.
    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd. Ed., Springer-Verlag, Berlin, 1999.CrossRefMATHGoogle Scholar
  21. 21.
    J.P. Serre, Cours d’arithmétique, PUF, 1970.Google Scholar
  22. 22.
    G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1944.MATHGoogle Scholar
  23. 23.
    D. Williams, Diffusions, Markov Processes and Martingales, vol. 1: Foundations, Wiley and Sons, New York, 1979.Google Scholar
  24. 24.
    M. Yor, Sur certaines fonctionnelles exponentielles du mouvement, brownien réel, J. Appl. Prob., 29 (1992), 202–208.MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    M. Yor, On some exponential functionals of Brownian motion, Adv. Appl. Prob., 24 (1992),.509–531.Google Scholar
  26. 26.
    M. Yor, From planar Brownian windings t,o Asian options, Insurance Math. Econom., 13 (1993), 23–34.MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    M. Yor (ed.), Exponential Functionals and Principal Values related to Brownian Motion, A collection of research papers, Biblioteca de la Revista Matemâtica Iberoamericana, 1997.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Catherine Donati-Martin
    • 1
  • Hiroyuki Matsumoto
    • 2
  • Marc Yor
    • 3
  1. 1.Laboratoire de Statistique et ProbabilitésUniversité Paul SabatierToulouse Cedex 04France
  2. 2.School of Informatics and SciencesNagoya UniversityChikusa-ku, NagoyaJapan
  3. 3.Laboratoire de ProbabilitésUniversité de Paris IVParisFrance

Personalised recommendations