Skip to main content
SpringerLink
Log in

The change of variables formula on Wiener space

  • Conference paper
  • First Online:
Séminaire de Probabilités XXXI

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1655))

Abstract

The transformations of measure induced by a not-necessarily adapted perturbation of the identity is considered. Previous results are reviewed and recent results on absolute continuity and related Radon-Nikodym densities are derived under conditions which are ‘as near as possible’ to the conditions of Federer’s area theorem in the finite dimensional case.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 65.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. De Gruyter Studies in Math., Vol. 14, Berlin, New York; De Gruyter 1991.

    Book  MATH  Google Scholar 

  2. Buckdahn, R.: Anticipative Girsanov transforms. Probab. Theory Relat. Fields 89, 211–238 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  3. Buckdahn, R.: Anticipative Girsanov transformations and Skorohod stochastic differential equations. Seminarbericht No. 92-2 (1992).

    Google Scholar 

  4. Cameron, R.H., Martin, W.T.: The transformation of Wiener integrals by nonlinear transformations. Trans. Amer. Math. Soc. 66, 253–283 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  5. Cruzeiro, A.B.: Equations differentielles sur l’espace de Wiener et formules de Cameron-Martin nonlinéaires. J. Functional Anal., Vol. 52, 335–347, (1984).

    Article  MathSciNet  Google Scholar 

  6. Enchev, O.: Nonlinear transformations on the Wiener space. The Annals of Probability, Vol. 21, 2169–2188 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  7. Enchev, O., D.W. Stroock: Anticipative diffusions and related change of measure. J. Functional Anal. 116, 449–477 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  8. Enchev, O., D.W. Stroock: Rademacher’s theorem for Wiener functionals. Annals of Probability 21 25–33 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  9. Federer, H.: Geometric Measure Theory, Springer-Verlag, 1969.

    Google Scholar 

  10. Kusuoka, S.: The nonlinear transformation of Gaussian measures on Banach space and its absolute continuity, I.J. Fac. Sci. Univ. Tokyo, Sect. IA, math. 29, 567–598 (1982).

    MathSciNet  MATH  Google Scholar 

  11. Nualart, D.: Markov fields and transformations of the Wiener measure. The Proceedings of the Oslo-Silivri Conference on stochastic analysis and related topics, p. 45–88, Gordon and Breach Stochastic Monograph, Vol. 8, T. Lindstrom, B. Oksendal and A. S. Üstünel (editors).

    Google Scholar 

  12. Nualart, D., Zakai, M.: Generalized stochastic integrals and the Malliavin calculus. Probab. Theory Relat. Fields 73, 255–280 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  13. Ramer, R.: On nonlinear transformations of Gaussian measures. J. Funct. Anal. 15, 166–187 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  14. Shepp, L.A.: Radon Nikodym derivatives of Gaussian measures. The Annals of Math. Stat. 37, 321–354 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  15. Üstünel, A.S.: Integrabilité exponentielle de fonctionnelles de Wiener. CRAS, Paris, Série I 305, 279–282 (1992).

    Google Scholar 

  16. Üstünel, A.S.: An Introduction to Analysis on Wiener Space. Lecture Notes in Math. Vol. 1610, Springer, 1995.

    Google Scholar 

  17. Üstünel, A.S., Zakai, M.: Transformations of Wiener measure under anticipative flows. Probab. Theory Relat. Fields 93, 91–136 (1992).

    Article  MATH  Google Scholar 

  18. Üstünel, A.S., Zakai, M.: Transformation of the Wiener measure under non-invertible shifts, Prob. Theory Related Fields, 99 485–500, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  19. Üstünel, A.S., Zakai, M.: Extensions of Lipschitz functions on Wiener space. To appear.

    Google Scholar 

  20. Üstünel, A.S., Zakai, M.: Measures induced on Wiener space by monotone shifts. To appear.

    Google Scholar 

  21. Üstünel, A.S., Zakai, M.: Applications of the degree theorem to absolute continuity on Wiener space. Probab. Theory Relat. Fields, 95, 509–520, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  22. Zakai, M., Zeitouni, O.: When does the Ramer formula look like the Girsanov formula? The Annals of Probability, Vol., 20, pp. 1436–1440, 1992.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dépt. Réseaux, ENST, 46 rue Barrault, 75013, Paris, France

    A. S. Üstünel

  2. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    M. Zakai

Authors
  1. A. S. Üstünel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Zakai
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jacques Azéma Marc Yor Michel Emery

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Üstünel, A.S., Zakai, M. (1997). The change of variables formula on Wiener space. In: Azéma, J., Yor, M., Emery, M. (eds) Séminaire de Probabilités XXXI. Lecture Notes in Mathematics, vol 1655. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0119289

Download citation

  • DOI: https://doi.org/10.1007/BFb0119289

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62634-3

  • Online ISBN: 978-3-540-68352-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation