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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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

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[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

A. S. Üstünel Parametric regularity of the conditional expectations via the Malliavin calculus and applications, Malliavin calculus and stochastic analysis. A Festschrift in honor of David Nualart, New York, NY: Springer, 2013, pp. 555-583 | DOI:10.1007/978-1-4614-5906-4_25 | Zbl:1401.60107
Rémi Lassalle Local invertibility of adapted shifts on Wiener space, under finite energy condition, Stochastics, Volume 85 (2013) no. 6, pp. 987-996 | DOI:10.1080/17442508.2012.720257 | Zbl:1290.60059
Mathias Beiglböck; Christian Léonard; Walter Schachermayer A generalized dual maximizer for the Monge–Kantorovich transport problem, ESAIM: Probability and Statistics, Volume 16 (2012), p. 306 | DOI:10.1051/ps/2011163
Rémi Lassalle Invertibility of adapted perturbations of the identity on abstract Wiener space, Journal of Functional Analysis, Volume 262 (2012) no. 6, pp. 2734-2776 | DOI:10.1016/j.jfa.2011.12.025 | Zbl:1251.60046

Cité par 4 documents. Sources : Crossref, zbMATH

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