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On some exponential functionals of Brownian motion

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Marc Yor
Marc Yor*
Affiliation:
Université Paris VI
*
∗Postal address: Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, Tour 56-3eme Etage, 75252 Paris Cedex 05, France.
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Abstract

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T] of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit-time distributions and the fixed time case is recovered by inverting Laplace transforms.

Keywords

BROWNIAN MOTION WITH DRIFTBESSEL FUNCTIONSBESSEL PROCESSESLAST-EXIT TIMES

MSC classification

Primary: 60J65: Brownian motion
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 3 , September 1992 , pp. 509 - 531
Copyright
Copyright © Applied Probability Trust 1992 

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