Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T19:26:15.857Z Has data issue: false hasContentIssue false
  • English
  • Français

On the integral of geometric Brownian motion

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Michael Schröder
Michael Schröder*
Affiliation:
Universität Mannheim
*
∗ Postal address: Keplerstrasse 30, D-69469 Weinheim, Germany. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the law of any real powers of the integral of geometric Brownian motion over finite time intervals. As its main results, an apparently new integral representation is derived and its interrelations with the integral representations for these laws originating by Yor and by Dufresne are established. In fact, our representation is found to furnish what seems to be a natural bridge between these other two representations. Our results are obtained by enhancing the Hartman-Watson Ansatz of Yor, based on Bessel processes and the Laplace transform, by complex analytic techniques. Systematizing this idea in order to overcome the limits of Yor's theory seems to be the main methodological contribution of the paper.

Keywords

Brownian motionLaplace transformAsian option

MSC classification

Primary: 60J65: Brownian motion
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 159 - 183
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alili, L., Matsumoto, H. and Shiraishi, T. (2001). On a triplet of exponential Brownian functionals. In Séminaire de Probabilités XXXV (Lecture Notes in Math. 1755), eds Azéma, J. et al., Springer, Berlin, pp. 396–415.Google Scholar
Doetsch, G. (1971). Handbuch der Laplace Transformation, Band I. Birkhäuser, Basel.Google Scholar
Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 1–2, 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. Osaka J. Math. 38, 379–381.Google Scholar
Dufresne, D. (2001). The integral of geometric Brownian motion. Adv. Appl. Prob. 33, 223–241.CrossRefGoogle Scholar
Lebedev, N. N. (1972). Special Functions and Their Applications. Dover, New York.Google Scholar
Rudin, W. (1987). Real and Complex Analysis, 3rd edn. McGraw-Hill, New York.Google Scholar
Schröder, M., (1997). On the valuation of Asian options: integral representations. Preprint, Universität Mannheim.Google Scholar
Schröder, M., (2002). Mathematical ramifications of option valuation: the case of the Asian option. Habilitationsschrift, Universität Mannheim.Google Scholar
Watson, G. N. (1944). A Treatise on The Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar
Whittaker, E. T. and Watson, G. N. (1965). A Course in Modern Analysis, 4th edn. Cambridge University Press.Google Scholar
Yor, M. (1980). Loi de l'indice du lacet brownien, et distribution de Hartman–Watson. Z. Wahrscheinlichkeitsth. 53, 71–95.CrossRefGoogle Scholar
Yor, M. (1992). On some exponential functionals of Brownian motion. Adv. Appl. Prob. 24, 509–531. (Reprinted as Chapter 2 in Y01.)Google Scholar
Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.Google Scholar
Yor, M. and Matsumoto, H. (2000). An analogue of Pitman's 2M-X theorem for exponential Wiener functionals. I. A time-inversion approach.. 159, 125–166.Google Scholar
Yor, M. and Matsumoto, H. (2001). A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration. Osaka J. Math. 38, 383–398.Google Scholar
Yor, M. and Matsumoto, H. (2003). On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts. To appear in Adv. Appl. Prob. 35, 184–206.Google Scholar
Yor, M. and Revuz, D. (1994). Continuous Martingales and Brownian Motion, 2nd edn. Springer, Berlin.Google Scholar
Yor, M., Ghomrasni, R. and Donati-Martin, C. (2001). On certain Markov processes attached to exponential functionals of Brownian motion; application to Asian options. Rev. Mat. Iberoamericana 17, 179–193.Google Scholar
Yor, M., Monthus, C. and Comtet., A. (1998). Exponential functionals of Brownian motion and disordered systems. J. Appl. Prob. 35, 255–271. (Reprinted as Chapter 8 in 15.)Google Scholar