no. 15-16
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

Üstünel, A. Suleyman1
1 Telecom-Paristech, Département Infres, 46, rue Barrault, 75013 Paris, France
Comptes Rendus. Mathématique, Tome 346 (2008) no. 15-16, pp. 897-900.

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

12∫W|u|H2dμ=∫WdUμdμlogdUμdμdμ.

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

12∫W|u|H2dμ=∫WdUμdμlogdUμdμdμ.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.07.013
Üstünel, A. Suleyman 1

1 Telecom-Paristech, Département Infres, 46, rue Barrault, 75013 Paris, France 
@article{CRMATH_2008__346_15-16_897_0,
     author = {\"Ust\"unel, A. Suleyman},
     title = {A necessary and sufficient condition for invertibility of adapted perturbations of identity on {Wiener} space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {897--900},
     publisher = {Elsevier},
     volume = {346},
     number = {15-16},
     year = {2008},
     doi = {10.1016/j.crma.2008.07.013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.07.013/}
}
TY  - JOUR
AU  - Üstünel, A. Suleyman
TI  - A necessary and sufficient condition for invertibility of adapted perturbations of identity on Wiener space
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 897
EP  - 900
VL  - 346
IS  - 15-16
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.07.013/
DO  - 10.1016/j.crma.2008.07.013
LA  - en
ID  - CRMATH_2008__346_15-16_897_0
ER  - 
%0 Journal Article
%A Üstünel, A. Suleyman
%T A necessary and sufficient condition for invertibility of adapted perturbations of identity on Wiener space
%J Comptes Rendus. Mathématique
%D 2008
%P 897-900
%V 346
%N 15-16
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.07.013/
%R 10.1016/j.crma.2008.07.013
%G en
%F CRMATH_2008__346_15-16_897_0
Üstünel, A. Suleyman. A necessary and sufficient condition for invertibility of adapted perturbations of identity on Wiener space. Comptes Rendus. Mathématique, Tome 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/

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

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

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

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

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

Cité par Sources :