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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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References

[1] Berthuet, R. (1986) Etude de processus généralisant l'aire de Lévy. Z. Wahrscheinlichkeitsth. 73, 463–480.CrossRefGoogle Scholar
[2] Chan, T., Dean, D., Jansons, K. and Rogers, C. (1992) Polymer conformations in elongational flows. Preprint.Google Scholar
[3] Chou, C. S. and Nualart, D. (1991) Sur l'extension d'une identité en loi entre le pont brownien et la variance du mouvement brownien. Preprint.Google Scholar
[4] Donati-Martin, C. and Yor, M. (1991) Fubini's theorem for double Wiener integrals and the variance of the Brownian path. Ann. Inst. H. Poincaré 27, 181–200.Google Scholar
[5] Donati-Martin, C., Song, S. and Yor, M. (1991) Symmetric stable processes and Fubini's theorem. Technical Report No. 314, Dept. of Statistics, University of California Berkeley.Google Scholar
[6] Foschini, G. J. and Shepp, L. A. (1991) Closed form characteristic functions for certain random variables related to Brownian motion. In Stochastic Analysis. Liber Amicorum for Moshe Zaka?, pp. 169–187. Academic Press, New York.CrossRefGoogle Scholar
[7] Helfer, A. and Zhongxin, Z. (1992) Gaussian integrals on Wiener spaces. J. Appl. Prob. 29, 46–55.CrossRefGoogle Scholar
[8] Pitman, J. and Yor, M. (1982) Sur une décomposition des ponts de Bessel. In Functional Analysis in Markov Processes, pp. 276–285, ed. Fukushima, M., Lecture Notes in Mathematics 923, Springer-Verlag, Berlin.Google Scholar
[9] Pitman, J. and Yor, M. (1982) A decomposition of Bessel bridges. Z. Wahrscheinlichkeitsth. 59, 425–457.Google Scholar
[10] Price, G., Rogers, C. and Williams, D. (1984). BM(R3) and its area integralß × dß. In Stochastic Analysis and Applications, Proceedings, Swansea 1983, ed. Truman, A. and Williams, D. Lecture Notes in Mathematics 1095, Springer-Verlag, New York.Google Scholar
[11] Williams, D. (1976) On a stopped Brownian motion formula of H. M. Taylor. In Séminaire de Probabilités X, pp. 235–239. Lecture Notes in Mathematics 511, Springer-Verlag, Berlin.Google Scholar
[12] Yor, M. (1980) Remarques sur une formule de P. Lévy. In Séminaire de Probabilités XIV, pp. 343–346. Lecture Notes in Mathematics 784, Springer-Verlag, Berlin.Google Scholar