Nagoya Mathematical Journal

An analogue of Pitman's {$2M-X$} theorem for exponential Wiener functionals. II. The role of the generalized inverse Gaussian laws

Hiroyuki Matsumoto and Marc Yor

Full-text: Open access


In Part I of this work, we have shown that the stochastic process $Z^{(\mu)}$ defined by (8.1) below is a diffusion process, which may be considered as an extension of Pitman's $2M-X$ theorem. In this Part II, we deduce from an identity in law partly due to Dufresne that $Z^{(\mu)}$ is intertwined with Brownian motion with drift $\mu$ and that the intertwining kernel may be expressed in terms of Generalized Inverse Gaussian laws.

Article information

Nagoya Math. J., Volume 162 (2001), 65-86.

First available in Project Euclid: 27 April 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J65: Brownian motion [See also 58J65]


Matsumoto, Hiroyuki; Yor, Marc. An analogue of Pitman's {$2M-X$} theorem for exponential Wiener functionals. II. The role of the generalized inverse Gaussian laws. Nagoya Math. J. 162 (2001), 65--86.

Export citation


  • L. Alili, D. Dufresne et M. Yor, Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement brownien avec drift , in [mono?].
  • O. Barndorff-Nielsen, Hyperbolic distributions and distributions on hyperbolae , Scand. J. Stat., 5 (1978), 151–157.
  • O. Barndorff-Nielsen, P. Blaesild and C. Halgreen, First hitting time models for the generalized inverse Gaussian distributions , Stoch. Proc. Appl., 7 (1978), 49–54.
  • O. Barndorff-Nielsen and N. Shephard, Aggregation and model construction for volatility models , preprint.
  • E. Bernadac, Fractions continues sur les matrices symétriques réelles et la loi Gaussienne inverse , C. R. Acad. Sci. Paris, Série I, 315 (1992), 329–332.
  • J. Bertoin and J. Pitman, Path transformations connecting Brownian bridge, excursion and meander , Bull. Sci. Math., 118 (1994), 147–166.
  • P. Biane and M. Yor, Quelques précisions sur le méandre brownien , Bull. Sci. Math., 112 (1988), 101–109.
  • D. Dufresne, The distribution of a perpetuity, with application to risk theory and pension funding , Scand. Actuarial J. (1990), 39–79.
  • ––––, An affine property of the reciprocal Asian option process , to appear in Osaka J. Math. 38 (2001).
  • I. J. Good, The population frequencies of species and the estimation of population parameters , Biometrika, 40 (1953), 237–264.
  • J.-P. Imhof, Density factorizations for Brownian motion, meander and the threedimensional Bessel process, and applications , J. Appl. Prob., 21 (1984), 500–510.
  • ––––, A simple proof of Pitman's $2M-X$ theorem , Adv. Appl. Prob., 24 (1992), 499–501.
  • N. N. Lebedev, Special Functions and their Applications, Dover, New York (1972).
  • G. Letac and V. Seshadri, A characterization of the generalized inverse Gaussian by continued fractions , Z. W., 62 (1983), 485–489.
  • G. Letac and J. Wesolowski, An independence property for the product of gig and gamma laws , Ann. Prob., 28 (2000), 1371–1383.
  • R. S. Liptser and A. N. Shiryayev, Statistics of Random Processes I, General Theory, Springer-Verlag, Berlin (1977).
  • H. Matsumoto and M. Yor, A relationship between Brownian motions with opposite drifts , to appear in Osaka J. Math. 38 (2001).
  • ––––, A version of Pitman's $2M-X$ theorem for geometric Brownian motions , C. R. Acad. Sc. Paris, Série I, 328 (1999), 1067–1074.
  • ––––, An analogue of Pitman's $2M-X$ theorem for exponential Wiener functionals, Part I : A time-inversion approach , Nagoya Math. J., 159 (2000), 125–166.
  • D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer-Verlag, Berlin (1999).
  • L. C. G. Rogers and J. W. Pitman, Markov functions , Ann. Prob., 9 (1981), 573–582.
  • V. Seshadri, The Inverse Gaussian Distributions, Oxford Univ. Press, Oxford (1993).
  • A. Terras, Harmonic Analysis on Symmetric Spaces and Applications II, Springer, Berlin (1988).
  • P. Vallois, La loi gaussienne inverse généralisée comme premier ou dernier temps de passage de diffusions , Bull. Sc. math., $2^e$ série, 115 (1991), 301–368.
  • G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge (1944).
  • D. Williams, Path decomposition and continuity of local time for one dimensional diffusions I , Proc. London Math. Soc., 28 (1974), 738–768.
  • M. Yor, On some exponential functionals of Brownian motion , Adv. Appl. Prob., 24 (1992), 509–531.
  • ––––, Sur certaines fonctionnelles exponentielles du mouvement brownien réel , J. Appl. Prob., 29 (1992), 202–208.
  • ––––, Tsirel'son's equation in discrete time , Prob. Th. Rel. Fields, 91 (1992), 135–152.
  • M. Yor (ed.), Exponential Functionals and Principal Values related to Brownian Motion, A collection of research papers, Biblioteca de la Revista Matemática Iberoamericana (1997).