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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: schroeder@math.uni-mannheim.de
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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 

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