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On Dufresne's relation between the probability laws of exponential functionals of Brownian motions with different drifts

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Hiroyuki Matsumoto  and
Marc Yor
Hiroyuki Matsumoto*
Affiliation:
Nagoya University
Marc Yor*
Affiliation:
Université Pierre et Marie Curie, Paris
*
∗ Postal address: Graduate School of Human Informatics, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan. Email address: matsu@info.human.nagoya-u.ac.jp
∗∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu, Case Courrier 188, F-75252 Paris Cedex 05, France.
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Abstract

Denote by αt(μ) the probability law of At(μ) = ∫0texp(2(Bs+μ s))ds for a Brownian motion {Bs, s ≥ 0}. It is well known that αt(μ) is of interest in a number of domains, e.g. mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for αt(0) and αt(1), as well as an equally remarkable relationship between αt(μ) and αt(ν) for two different drifts μ and ν. In this paper, hinging on previous results about αt(μ), we give different proofs of Dufresne's results and present extensions of them for the processes {At(μ), t ≥ 0}.

Keywords

Brownian motions with driftsBrownian exponential functionalsdensities in closed form

MSC classification

Primary: 60J65: Brownian motion 60J25: Continuous-time Markov processes on general state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 184 - 206
Copyright
Copyright © Applied Probability Trust 2003 

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