Advertisement

On certain probabilities equivalent to Wiener measure, d'Après Dubins, Feldman, Smorodinsky and Tsirelson

  • W. Schachermayer
Questions de Filtrations
Part of the Lecture Notes in Mathematics book series (LNM, volume 1709)

Abstract

L. Dubins, J. Feldman, M. Smorodinsky and B. Tsirelson gave an example of an equivalent measure Q on standard Wiener space such that each adapted Q-Brownian motion generates a strictly smaller filtration then the original one. The construction of this important example is complicated and technical.

We give a variant of their construction which differs in some of the technicalities but essentially follows their ideas, hoping that some readers may find our presentation easier to digest than the original papers.

Keywords

Brownian Motion Generate Parametrisation Polish Space Borel Function Equivalent Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BEKSY 98] M.T. Barlow, M. Émery, F.B. Knight, S. Song, M. Yor, Autour d'un théorème de Tsirelson sur des filtrations Browniennes et non Browniennes, Séminaire de Probabilités XXXII, LNM, Springer 1686 (1998), 264–305.Google Scholar
  2. [BE 99] S. Beghdadi-Sakrani, M. Émergy, On certain probabilities equivalent to cointossing, d'après Schachermayer, in this volume (1999).Google Scholar
  3. [DFST 96] L. Dubins, J. Feldman, M. Smorodinsky, B. Tsirelson, Decreasing Sequences of σ-fields and a Measure Change for Brownian Motion, Annals of Probability 24 (1996), 882–904.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [EY 98] M. Émery, M. Yor, Sur un théorème de Tsirelson relatif à des mouvements Browniens corrélés et à la nullité de certain temps locaux, Séminaire de Probabilités XXXII, LNM, Springer 1686 (1998), 306–312.Google Scholar
  5. [F 96] J. Feldman, ε-close Measures Producing Nonisomorphic Filtrations, Annals of Probability 24 (1996), 912–914.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [FT 96] J. Feldman, B. Tsirelson, Decreasing Sequences of σ-fields and a Measure Change for Borwnian Motion II, Annals of Probability 24 (1996), 905–911.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [RY 91] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (1991).Google Scholar
  8. [RY 94] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, second edition, Springer (1994).Google Scholar
  9. [S 98] M. Smorodinsky, An Example of a non Sub-Standard Process, to appear in Israel Journal of Math. (1998).Google Scholar
  10. [SY 80] D.W. Stroock, M. Yor, On extremal solutions of martingale problems, Ann. Sci. Ecole Norm. Sup. 13 (1980), 95–164.MathSciNetzbMATHGoogle Scholar
  11. [SY 81] D.W. Stroock, M. Yor, Some remarkable martingales, Sém. Prob. XV, LNM 850 (1981), 590–603.MathSciNetzbMATHGoogle Scholar
  12. [T 97] B. Tsirelson, Triple points: from non-Brownian filtrations to harmonic measures, GAFA Geometric And Functional Analysis, Birkhäuser 7 (1997), 1096–1142.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [V 70] A.M. Vershik, Decreasing sequences of measurable partitions and their applications, Soviet. Math. Dokl. 11 (1970), 1007–1011.zbMATHGoogle Scholar
  14. [V 73] A.M. Vershik, Approximation in measure theory. Dissertation, Leningrad Univ., In Russian (1973).Google Scholar
  15. [V 95] A.M. Vershik, The Theory of Decreasing Sequences of Measurable Partitions, St. Petersburg Math. Journal 6/4 (1995), 705–761. *** DIRECT SUPPORT *** A00I6C60 00005MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • W. Schachermayer

There are no affiliations available

Personalised recommendations