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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: [email protected]
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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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