Probability Theory
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.

Let (W,H,μ) be the classical Wiener space, assume that U=IW+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

12W|u|H2dμ=WdUμdμlogdUμdμdμ.

Soit (W,H,μ) l'espace de Wiener classique, et soit U=IW+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

12W|u|H2dμ=WdUμdμlogdUμdμdμ.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.07.013
Üstünel, A. Suleyman 1

1 Telecom-Paristech, Département Infres, 46, rue Barrault, 75013 Paris, France
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Üstünel, A. Suleyman. 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. http://www.numdam.org/articles/10.1016/j.crma.2008.07.013/

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