A necessary and sufficient condition for invertibility of adapted perturbations of identity on Wiener space

A. Suleyman Üstünel

doi:10.1016/j.crma.2008.07.013

Probability Theory

A necessary and sufficient condition for invertibility of adapted perturbations of identity on Wiener space
[Une condition nécessaire et suffisante pour l'inversibilité de perturbations d'identité adaptées sur l'espace de Wiener]

Présenté par : Paul Malliavin

A. Suleyman Üstünel ¹

¹ Telecom-Paristech, Département Infres, 46, rue Barrault, 75013 Paris, France

Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 897-900.

Résumé
Abstract

Soit $(W, H, μ)$ l'espace de Wiener classique, et soit $U = I_{W} + u$ une perturbation d'identité adaptée satisfaisant á l'identité de Girsanov. Alors U est inversible si et seulement si l'énergie cinétique de u est égale à l'entropie relative de la mesure induite par l'action de U sur la mesure de Wiener μ ; en d'autres termes U est inversible si et seulement si

\frac{1}{2} \int_{W} {| u |}_{H}^{2} d μ = \int_{W} \frac{d U μ}{d μ} \log \frac{d U μ}{d μ} d μ .

Let $(W, H, μ)$ be the classical Wiener space, assume that $U = I_{W} + u$ is an adapted perturbation of identity satisfying the Girsanov identity. Then, U is invertible if and only if the kinetic energy of u is equal to the relative entropy of the measure induced with the action of U on the Wiener measure μ, in other words U is invertible if and only if

\frac{1}{2} \int_{W} {| u |}_{H}^{2} d μ = \int_{W} \frac{d U μ}{d μ} \log \frac{d U μ}{d μ} d μ .

Reçu le : 2008-07-07
Accepté le : 2008-07-10
Publié le : 2008-08-01

DOI : 10.1016/j.crma.2008.07.013

Affiliations des auteurs :

A. Suleyman Üstünel ¹

¹ Telecom-Paristech, Département Infres, 46, rue Barrault, 75013 Paris, France

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     author = {A. Suleyman \"Ust\"unel},
     title = {A necessary and sufficient condition for invertibility of adapted perturbations of identity on {Wiener} space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {897--900},
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     doi = {10.1016/j.crma.2008.07.013},
     language = {en},
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A. Suleyman Üstünel. A necessary and sufficient condition for invertibility of adapted perturbations of identity on Wiener space. Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 897-900. doi : 10.1016/j.crma.2008.07.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.07.013/

Bibliographie
Cité par

[1] D. Feyel; A.S. Üstünel; M. Zakai Realization of positive random variables via absolutely continuous transformations of measure on Wiener space, Probability Surveys, Volume 3 (2006), pp. 170-205 (electronic)

[2] M. Fujisaki; G. Kallianpur; H. Kunita Stochastic differential equations for the non linear filtering problem, Osaka J. Math., Volume 9 (1972), pp. 19-40

[3] N. Ikeda; S. Watanabe Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981 (Kodansha Ltd., Tokyo)

[4] A.S. Üstünel; M. Zakai Transformation of Measure on Wiener Space, Springer-Verlag, 1999

[5] A.S. Üstünel; M. Zakai The invertibility of adapted perturbations of identity on the Wiener space, C. R. Acad. Sci. Paris, Ser. I, Volume 342 (2006), pp. 689-692

[6] A.S. Üstünel; M. Zakai Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space, Probab. Theory Relat. Fields, Volume 139 (2007), pp. 207-234

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