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An exponential functional of random walks

Part of: Limit theorems Stochastic processes Applications

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.
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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 

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Footnotes

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

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