Exponential functionals of Brownian motion, I: Probability laws at fixed time

This paper is the rst part of our survey on various results about the distribution of exponential type Brownian functionals dened as an integral over time of geometric Brownian motion. Several related topics are also mentioned.

[1]  J. Lamperti Semi-stable Markov processes. I , 1972 .

[2]  Marc Yor,et al.  Exponential functionals and principal values related to brownian motion : a collection of research papers , 1997 .

[3]  M. Yor,et al.  Some Absolute Continuity Relationships for Certain Anticipative Transformations of Geometric Brownian Motions , 2001 .

[4]  M. Yor,et al.  Exponential functionals of Brownian motion and disordered systems , 1996, Journal of Applied Probability.

[5]  S. Taylor DIFFUSION PROCESSES AND THEIR SAMPLE PATHS , 1967 .

[6]  Some consequences of the cyclic exchangeability property for exponential functionals of Lévy processes , 2001 .

[7]  G. Oshanin,et al.  Steady flux in a continuous-space Sinai chain , 1993 .

[8]  J. Paulsen,et al.  On the distribution of a randomly discounted compound Poisson process , 1996 .

[9]  S. Majumdar,et al.  Exact asymptotic results for persistence in the Sinai model with arbitrary drift. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  M. Yor Some Aspects Of Brownian Motion , 1992 .

[11]  Geoffrey S. Watson,et al.  "Normal" Distribution Functions on Spheres and the Modified Bessel Functions , 1974 .

[12]  M. Yor,et al.  Exponential functionals of Levy processes , 2005, math/0511265.

[13]  Marc Yor,et al.  On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes , 2002 .

[14]  M. Yor,et al.  Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson , 1980 .

[15]  Jostein Paulsen,et al.  Present value distributions with applications to ruin theory and stochastic equations , 1997 .

[16]  M. Yor,et al.  On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts , 2003, Advances in Applied Probability.

[17]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[18]  M. Yor,et al.  Some Properties of the Wishart Processes and a Matrix Extension of the Hartman-Watson Laws † , 2004 .

[19]  M. Yor,et al.  BESSEL PROCESSES, ASIAN OPTIONS, AND PERPETUITIES , 1993 .

[20]  C. Monthus Etude de quelques fonctionnelles du mouvement brownien et de certaines propriétés de la diffusion unidimensionnelle en milieu aléatoire , 1995 .

[21]  P. Salminen,et al.  Properties of perpetual integral functionals of Brownian motion with drift , 2005 .

[22]  H. Bateman,et al.  Higher Transcendental Functions [Volumes I-III] , 1953 .

[23]  M. Yor,et al.  A study of the Hartman-Watson distribution motivated by numerical problems related to the pricing of Asian options , 2004 .

[24]  P. Bougerol Exemples de théorèmes locaux sur les groupes résolubles , 1983 .

[25]  Marc Yor,et al.  An analogue of Pitman’s 2M – X theorem for exponential Wiener functionals: Part I: A time-inversion approach , 2000, Nagoya Mathematical Journal.

[26]  M. Yor Sur certaines fonctionnelles exponentielles du mouvement brownien réel , 1992, Journal of Applied Probability.

[27]  R. Bhattacharya,et al.  A note on the distribution of integrals of geometric Brownian motion , 2001 .

[28]  Analytic approach to Yor’s formula of exponential additive functionals of Brownian motion , 1996 .

[29]  Sur l'identité de Bougerol pour les fonctionnelles exponentielles du mouvement brownien avec drift , 1997 .

[30]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[31]  D. Dufresne Laguerre Series for Asian and Other Options , 2000 .

[32]  Analysis on root systems , 1987 .

[33]  M. Yor On some exponential functionals of Brownian motion , 1992, Advances in Applied Probability.

[34]  A. Comtet,et al.  Diffusion in a one-dimensional random medium and hyperbolic Brownian motion , 1995, cond-mat/9506024.

[35]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[36]  A. Borodin,et al.  Some Exponential Integral Functionals of BM(μ) and BES(3) , 2006 .

[37]  J. Gruet Semi-groupe du mouvement brownienhyperbolique , 1996 .

[38]  Jostein Paulsen,et al.  Risk theory in a stochastic economic environment , 1993 .

[39]  C. Grosche,et al.  Handbook of Feynman Path Integrals , 1995 .

[40]  Kazuyuki Ishiyama Methods for Evaluating Density Functions of Exponential Functionals Represented as Integrals of Geometric Brownian Motion , 2005 .

[41]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[42]  Marc Yor,et al.  Some aspects of Brownian motion: part i: some special functionals , 1992 .

[43]  N. Ikeda,et al.  Brownian Motion on the Hyperbolic Plane and Selberg Trace Formula , 1999 .

[44]  Sur la convergence absolue de certaines integrales , 1982 .

[45]  M. Yor Exponential Functionals of Brownian Motion and Related Processes , 2001 .

[46]  T. Stieltjes Recherches sur les fractions continues , 1995 .

[47]  Marc Yor,et al.  The Entrance Laws of Self-Similar Markov Processes and Exponential Functionals of Lévy Processes , 2002 .

[48]  ON BOUGEROL AND DUFRESNE'S IDENTITIES FOR EXPONENTIAL BROWNIAN FUNCTIONALS , 1998 .

[49]  D. Dufresne The Distribution of a Perpetuity, with Applications to Risk Theory and Pension Funding , 1990 .

[50]  Daniel Dufresne The integral of geometric Brownian motion , 2001 .

[51]  G. A. Watson A treatise on the theory of Bessel functions , 1944 .

[52]  M. Yor,et al.  On positive and negative moments of the integral of geometric Brownian motions , 2000 .

[53]  M. Yor,et al.  Limiting distributions associated with moments of exponential Brownian functionals , 2004 .

[54]  M. Yor Sur les lois des fonctionnelles exponentielles du mouvement brownin, considérées en certains instants aléatories , 1992 .

[55]  B. Leblanc Une approche unifiée pour une forme exacte du prix d'une option dans les différents modèles a volatilité stochastique , 1996 .

[56]  L. Alili,et al.  On a triplet of exponential Brownian functionals , 2001 .

[57]  Yuu Hariya A time-change approach to Kotani's extension of Yor's formula , 2005 .

[58]  A. Basu,et al.  The Inverse Gaussian Distribution , 1993 .

[59]  H. Matsumoto Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one , 2001 .

[60]  M. Yor,et al.  The Law of Geometric Brownian Motion and its Integral, Revisited; Application to Conditional Moments , 2002 .

[61]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[62]  H. McKean,et al.  Diffusion processes and their sample paths , 1996 .

[63]  H. Buchholz The Confluent Hypergeometric Function , 2021, A Course of Modern Analysis.