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Exponential functionals of Brownian motion and disordered systems

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Alain Comtet ,
Cécile Monthus  and
Marc Yor
Alain Comtet*
Affiliation:
IPN
Cécile Monthus*
Affiliation:
C.E. Saclay
Marc Yor*
Affiliation:
Université Paris 6
*
∗Postal address: (1) Division de Physique Théorique (Unité de Recherche des Universités Paris 6 et Paris 11 associée au CNRS), IPN Bâtiment 100, 91406 Orsay Cédex; (2) L.P.T.P.E., Tour 12, 4 Place Jussieu 75252 Paris Cedex 05, France. E-mail address: [email protected]
∗∗Postal address: Service de Physique Théorique, C. E. Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette Cédex, France. E-mail address: [email protected]
∗∗∗Postal address: Laboratoire de Probabilités, Université Paris 6, 4 Place Jussieu, Tour 56, 75252 Paris Cedex 05, France.
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Abstract

The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts, such as continuous time finance models and one-dimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.

Keywords

Exponentials of Brownian motiondisordered systemspartition functionfree energy

MSC classification

Primary: 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 255 - 271
Copyright
Copyright © Applied Probability Trust 1998 

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References

Alili, L. (1995). Fonctionnelles exponentielles et valeurs principales du mouvement Brownien. Thèse de l'Université Paris 6.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 Exponential Functionals and Principal Values related to Brownian Motion. A collection of Biblioteca de la Revista Matematica research papers, ed. Yor, M. Ibero–Americana, p. 3.Google Scholar
Biane, P., and Yor, M. (1987). Valeurs principales associées aux temps locaux Browniens. Bull. Sci. Math. 111, 23–101.Google Scholar
Bouchaud, J.P., Comtet, A., Georges, A., and Le Doussal, P. (1990). Classical diffusion of a particle in a one-dimensional random force field. Ann. Phys. 201, 285–341.Google Scholar
Bouchaud, J.P., and Georges, A. (1990). Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293.Google Scholar
Bougerol, P. (1983). Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré 19, 369–391.Google Scholar
Broderix, K., and Kree, R. (1995). Thermal equilibrium with the Wiener potential: testing the Replica variational approximation. Europhys. Lett. 32, 343–348.Google Scholar
Burlatsky, S.F., Oshanin, G.H., Mogutov, A.V., and Moreau, M. (1992). Non-Fickian steady flux in a one-dimensional Sinaï-type disordered system. Phys. Rev. A 45, 6955–6957.CrossRefGoogle Scholar
Carmona, Ph., Petit, F., and Yor, M. (1994). Sur les fonctionnelles exponentielles de certains processus de Lévy. Stochast. Rep. 47, p. 71.Google Scholar
Carmona, Ph., Petit, F., and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values related to Brownian Motion. A collection of Biblioteca de la Revista Matematica research papers, ed. Yor, M. Ibero–Americana, p. 73.Google Scholar
Comtet, A., and Monthus, C. (1996). Diffusion in one-dimensional random medium and hyperbolic Brownian motion. J. Phys. A: Math. Gen. 29, 1331–1345.Google Scholar
Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Act. J. 39.Google Scholar
Gardner, E., and Derrida, B. (1989). The probability distribution of the partition function of the Random Energy Model. J. Phys. A: Math. Gen. 22, 1975–1981.Google Scholar
Geman, H., and Yor, M. (1993). Bessel processes, Asian options and perpetuities. Math. Fin. 3, 349–375.Google Scholar
Georges, A. (1988). Diffusion anormale dans les milieux désordonnés: Mécanismes statistiques, modèles théoriques et applications. Thèse d'état de l'Université Paris 11.Google Scholar
Hongler, M.O., and Desai, R.C. (1986). Decay of unstables states in the presence of fluctuations. Helv. Phys. Acta 59, 367–389.Google Scholar
Kawazu, K., and Tanaka, H. (1993). On the maximum of a diffusion process in a drifted Brownian environment. Sem. Prob. XXVII, 78 Lect. Notes in Math. Springer, p. 1557.Google Scholar
Kent, J. (1978). Some probabilistic properties of Bessel functions. Ann. Prob. 6, p. 760.CrossRefGoogle Scholar
Kesten, H., Koslov, M., and Spitzer, F. (1975). A limit law for random walk in a random environment. Compositio Math. 30, p. 145.Google Scholar
Lebedev, N. (1972). Special Functions and their Applications. Dover.Google Scholar
Monthus, C., and Comtet, A. (1994). On the flux in a one-dimensional disordered system. J. Phys. I (France) 4, 635–653.Google Scholar
Monthus, C., Oshanin, G., Comtet, A., and Burlatsky, S.F. (1996). Sample-size dependence of the ground-state energy in a one-dimensional localization problem. Phys. Rev. E 54, 231–242.Google Scholar
Opper, M. (1993). Exact solution to a toy random field model. J. Phys. A: Math. Gen. 26, L719–L722.CrossRefGoogle Scholar
Oshanin, G., Mogutov, A., and Moreau, M. (1993). Steady flux in a continuous Sinaï chain. J. Stat. Phys. 73, 379–388.Google Scholar
Oshanin, G., Burlatsky, S.F., Moreau, M., and Gaveau, B. (1993). Behavior of transport characteristics in several one-dimensional disordered systems. Chem. Phys. 177, 803–819.Google Scholar
Pitman, J., and Yor, M. (1993a). A limit theorem for one-dimensional Brownian motion near its maximum, and its relation to a representation of the two-dimensional Bessel bridges. Preprint.Google Scholar
Pitman, J., and Yor, M. (1993b). Dilatations d'espace-temps, réarrangement des trajectoires Browniennes, et quelques extensions d'une identité de Knight. C. R. Acad. Sci. Paris 316, 723–726 Google Scholar
Pitman, J., and Yor, M. (1996). Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. In Itô's Stocahastic Calculus and Probability Theory, eds. Ikeda, N., Watanabe, S., Fukushima, M. and Kunita, H. Springer, p. 293. %To appear in a %Festschrift volume in honor of K. Itô, Springer.Google Scholar
de Schepper, A., Goovaerts, M., and Delbaen, F. (1992). The Laplace transform of annuities certain with exponential time distribution. Ins. Math. Econ. 11, p. 291.Google Scholar
Wong, E. (1964). The construction of a class of stationary Markov processes. In Am. Math. Soc. Proc. of the 16th Symposium of Appl. Math. p. 264.Google Scholar
Yor, M. (1992). On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509–531.Google Scholar
Yor, M. (1992). Sur les lois des fonctionnelles exponentielles du mouvement Brownien, considérées en certains instants aléatoires. C. R. Acad. Sci. Paris 314, 951–956.Google Scholar
Yor, M. (1992). Some aspects of Brownian motion. Part I: Some special functionals. Lectures in Mathematics. ETH Zürich, Birkhaüser.Google Scholar
Yor, M. (1993). Sur une fonctionnelle exponentielle du mouvement Brownien réel. J. Appl. Prob. 29, 202–209.Google Scholar
Yor, M. (1993). From planar Brownian windings to Asian options. Ins. Math. Econ. 13.Google Scholar