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On some examples of quadratic functionals of Brownian motion

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

C. Donati-Martin  and
M. Yor
C. Donati-Martin*
Affiliation:
Université de Provence
M. Yor*
Affiliation:
Université Pierre et Marie Curie
*
∗ Postal address: Université de Provence, U.R.A. 225, 3 Place Victor Hugo, F13331 Marseille Cédex 3, France.
∗∗ Postal address: Laboratoire de Probabilités, Université P. et M. Curie, 4 Place Jussieu, F75252 Paris Cédex 05, France.
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Abstract

During the last few years, several variants of P. Lévy's formula for the stochastic area of complex Brownian motion have been obtained. These are of interest in various domains of applied probability, particularly in relation to polymer studies. The method used by most authors is the diagonalization procedure of Paul Lévy.

Here we derive one such variant of Lévy's formula, due to Chan, Dean, Jansons and Rogers, via a change of probability method, which reduces the computation of Laplace transforms of Brownian quadratic functionals to the computations of the means and variances of some adequate Gaussian variables.

We then show that with the help of linear algebra and invariance properties of the distribution of Brownian motion, we are able to derive simply three other variants of Lévy's formula.

Keywords

LÉVYS FORMULAGIRSANOV'S THEOREMRADIUS OF GYRATIONCORRELATED BROWNIAN MOTIONS

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60J60: Diffusion processes
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 570 - 584
Copyright
Copyright © Applied Probability Trust 1993 

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