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.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
[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.
[BE 99] S. Beghdadi-Sakrani, M. Émergy, On certain probabilities equivalent to cointossing, d'après Schachermayer, in this volume (1999).
[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.
[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.
[F 96] J. Feldman, ε-close Measures Producing Nonisomorphic Filtrations, Annals of Probability 24 (1996), 912–914.
[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.
[RY 91] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer (1991).
[RY 94] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, second edition, Springer (1994).
[S 98] M. Smorodinsky, An Example of a non Sub-Standard Process, to appear in Israel Journal of Math. (1998).
[SY 80] D.W. Stroock, M. Yor, On extremal solutions of martingale problems, Ann. Sci. Ecole Norm. Sup. 13 (1980), 95–164.
[SY 81] D.W. Stroock, M. Yor, Some remarkable martingales, Sém. Prob. XV, LNM 850 (1981), 590–603.
[T 97] B. Tsirelson, Triple points: from non-Brownian filtrations to harmonic measures, GAFA Geometric And Functional Analysis, Birkhäuser 7 (1997), 1096–1142.
[V 70] A.M. Vershik, Decreasing sequences of measurable partitions and their applications, Soviet. Math. Dokl. 11 (1970), 1007–1011.
[V 73] A.M. Vershik, Approximation in measure theory. Dissertation, Leningrad Univ., In Russian (1973).
[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 00005
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag
About this paper
Cite this paper
Schachermayer, W. (1999). On certain probabilities equivalent to Wiener measure, d'Après Dubins, Feldman, Smorodinsky and Tsirelson. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXIII. Lecture Notes in Mathematics, vol 1709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096513
Download citation
DOI: https://doi.org/10.1007/BFb0096513
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66342-3
Online ISBN: 978-3-540-48407-3
eBook Packages: Springer Book Archive
