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The integral of geometric Brownian motion

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Daniel Dufresne
Daniel Dufresne*
Affiliation:
Université de Montréal
*
∗ Postal address: Département de mathématiques et de statistique, Université de Montréal, PO Box 6128, Downtown Station, Montréal, Québec, Canada H3C 3J7. Email address: dufresne@dms.umontreal.ca
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Abstract

This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

Keywords

Brownian motionBougerol's identityAsian options

MSC classification

Primary: 60J65: Brownian motion
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 223 - 241
Copyright
Copyright © Applied Probability Trust 2001 

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References

Alili, L. (1995). Fonctionnelles exponentielles et valeurs principales du mouvement brownien. , Université Paris VI.Google Scholar
Alili, L., Dufresne, D. and Yor, M. (1997). Sur l'identité de Bougerol pour les fonctionelles exponentielles du mouvement brownien avec drift. In Exponential Functionals and Principal Values Related to Brownian Motion, ed. Yor, M.. Revista Matemática Iberoamericana, Madrid, pp. 3–14.Google Scholar
Bougerol, P. (1983). Exemples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré 19, 369–391.Google Scholar
Carmona, P., 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, ed. Yor, M.. Revista Matemática Iberoamericana, Madrid, pp. 73–121.Google Scholar
Comtet, A. and Monthus, C. (1996). Diffusion in one-dimensional random medium and hyperbolic Brownian motion. J. Phys. A. 29, 1331–1345.CrossRefGoogle Scholar
Comtet, A., Monthus, C. and Yor, M. (1998). Exponential functionals of Brownian motion and disordered systems. J. Appl. Prob. 35, 255–271.CrossRefGoogle Scholar
De Schepper, A., Goovaerts, M. and Delbaen, F. (1992). The Laplace transform of annuities certain with exponential time distribution. Insurance: Math. Econ. 11, 291–294.Google Scholar
Donati-Martin, C., Ghomrasni, R. and Yor, M. (2001). On certain Markov processes attached to exponential functionals of Brownian motion. To appear in Rev. Mat. Iberoamericana.Google Scholar
Dufresne, D. (1989). Weak convergence of random growth processes with applications to insurance. Insurance: Math. Econ. 8, 187–201.Google Scholar
Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial. J. 1990, 39–79.CrossRefGoogle Scholar
Dufresne, D. (2000). Laguerre series for Asian and other options. Math. Finance 10, 407–428.CrossRefGoogle Scholar
Dufresne, D. (2001). An affine property of the reciprocal Asian option process. To appear in Osaka J. Math. Google Scholar
Lebedev, N. N. (1972). Special Functions and their Applications. Dover, New York.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 Ser. I 328, 1067–1074.CrossRefGoogle Scholar
Matsumoto, H. and Yor, M. (2001). A relationship between Brownian motions with opposite drifts via certain enlargments of the Brownian filtration. To appear in Osaka J. Math.Google Scholar
Monthus, C. and Comtet, A. (1994). On the flux in a one-dimensional disordered system. J. Physique I 4, 635–653.CrossRefGoogle Scholar
Widder, D. V. (1946). The Laplace Transform. Princeton University Press.Google Scholar
Yor, M. (1992a). On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509–531.CrossRefGoogle Scholar
Yor, M. (1992b). Sur certaines fonctionnelles du mouvement brownien réel. J. Appl. Prob. 29, 202–208.CrossRefGoogle Scholar
Yor, M. (1992c). Sur les lois des fonctionnelles exponentielles du mouvement brownien, considérées en certains instants aléatoires. C. R. Acad. Sci. Paris I 314, 951–956.Google Scholar