Summary
In this paper we study some classes of Wiener functionals whose elements can be composed with a non-linear, non-absolutely continous transformation of the form of perturbation of identity in the direction of Cameron-Martin space. We show that under certain conditions the image of the Wiener measure under the above transformation induces a generalized Wiener functional on certain Sobolev spaces generalizing the Radon-Nikodym relation to non absolutely continuous transformations. A series representation for the generalized Radon-Nikodym derivative is presented and conditional expectations of some generalized random variables are considered.
References
Bismut, J.-M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander conditions. Z. Wahrescheinlichkeitstheor. Verw. Geb.56, 469–505 (1981)
Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space. Berlin New York: de Gruyter 1991
Korezlioglu, H., Üstünel, A.S.: Distributions, Girsanov and degree theorems on Wiener space. In: Hida, T., Kuo, H.H., Potthoff, J., Streit, L. (eds.) White Noise Analysis, pp. 231–245. Singapore: World Scientific 1990
Kusuoka, S.: The nonlinear transformation of Gaussian measure on Banach spaces and its absolute continuity. J. Fac. Sci. Univ. Tokyo, Sect. 1 A, Math.29, 572 (1982)
Kuo, H.-H.: The chain rule for differentiable measures. Stud. Math.LXIII, 146–155 (1978)
Nualart, D., Üstünel, A.S., Zakai, M.: Some relations among classes of σ-fields on Wiener space. Probab. Theory Relat. Fields85, 119–129 (1990)
Nualart, D., Zakai, M.: A summary of some identities of the Malliavin calculus. In: Da Prato, G., Tubaro, L. (eds.) Stochastic partial differential equations and applications II. (Lect. Notes Math., vol. 1390, pp. 192–196) Berlin Heidelberg New York: Springer 1989
Nualart, D., Zakai, M.: Positive and strongly positive Wiener functionals. (to appear)
Perez-Abreu, V.: Anticipative solutions of stochastic bilinear equations in Hilbert spaces. In: Gorostiza, L., Leon, J.R. (eds.) Proceedings CLAPEM, pp. 297–312. Mexico 1992
Stroock, D.: Homogeneous chaos revisited. In: Azema, J., Meyer, P.A., Yor, M. (eds.) Seminaire de probabilities XXI. (Lect. Notes Math., vol. 1321, pp. 1–7) Berlin Heidelberg New York: Springer 1988
Üstünel, A.S.: Intégrabilité exponentielle de fonctionnelles de Wiener. C.R. Acad. Sci. Paris, Sér. I315, 997–1000 (1992)
Üstünel, A.S.: Some applications of stochastic integration in infinite dimensions. Stochastics7, 225–288 (1982)
Watanabe, S.: Lectures on Stochastic differential equations and the Malliavin calculus. Tata Institute of Fundamental Research. Berlin Heidelberg New York: Springer 1984
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Üstünel, A.S., Zakai, M. The composition of Wiener functionals with non absolutely continuous Shifts. Probab. Th. Rel. Fields 98, 163–184 (1994). https://doi.org/10.1007/BF01192512
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01192512
Mathematics Subject Classifiation
- 60H05
- 60G15
- 60G30
- 46G05
- 46G12
- 58B10