Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T05:41:26.338Z Has data issue: false hasContentIssue false
  • English
  • Français

An exponential functional of random walks

Part of: Limit theorems Applications Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Tamás Szabados  and
Balázs Székely
Tamás Szabados*
Affiliation:
Budapest University of Technology and Economics
Balázs Székely*
Affiliation:
Budapest University of Technology and Economics
*
∗ Postal address: Department of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3, H ép. V em., Budapest, 1521, Hungary.
∗ Postal address: Department of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3, H ép. V em., Budapest, 1521, Hungary.
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

Keywords

Exponential functionalAsian optionrandom walkstrong approximationself-similarityLp limit

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles 60F25: $L^p$-limit theorems
Secondary: 62P05: Applications to actuarial sciences and financial mathematics 60G50: Sums of independent random variables; random walks
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 2 , June 2003 , pp. 413 - 426
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.)

Footnotes

Research supported by the French—Hungarian intergovermental grant ‘Balaton’ F-39/200.

References

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., Biblioteca de la Revista Matemática Iberoamericana, Madrid, pp. 73–121.Google Scholar
Csörgő, M. (1999). Random walking around financial mathematics. In Random Walks (Bolyai Soc. Math. Stud. 9), eds Révész, P. and Tóth, B., János Bolyai Mathematical Society, Budapest, pp. 59–111.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.Google Scholar
Falconer, K. (1990). Fractal Geometry. Mathematical Foundations and Applications. John Wiley, Chichester.Google Scholar
Grincevivcius, A. K. (1974). On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines. Theory Prob. Appl. 19, 163–168.CrossRefGoogle Scholar
Knight, F. B. (1962). On the random walk and Brownian motion. Trans. Amer. Math. Soc. 103, 218–228.Google Scholar
Révész, P. (1990). Random Walk in Random and Nonrandom Environments. World Scientific, Singapore.CrossRefGoogle Scholar
Rudin, W. (1970). Real and Complex Analysis. McGraw-Hill, New York.Google Scholar
Simon, B. (1998). The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203.Google Scholar
Simon, K., Solomyak, B. and Urbański, M. (2001). Invariant measures for parabolic IFS with overlaps and random continued fractions. Trans. Amer. Math. Soc. 353, 5145–5164.Google Scholar
Szabados, T. (1996). An elementary introduction to the Wiener process and stochastic integrals. Studia Sci. Math. Hung. 31, 249–297.Google Scholar
Székely, G. J. (1975). On the polynomials of independent random variables. In Limit Theorems of Probability Theory (Colloq. Math. Soc. János Bolyai 11), North-Holland, Amsterdam, pp. 365–371.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar
Yor, M. (1992). On certain exponential functionals of real-valued Brownian motion. J. Appl. Prob. 29, 202–208.Google Scholar
Yor, M. (2001). Exponential Functionals of Brownian Motion and Related Processes. Springer, Berlin.Google Scholar