Skip to main content

Skew-Product Decomposition of Planar Brownian Motion and Complementability

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2123)

Abstract

Let Z be a complex Brownian motion starting at 0 and W the complex Brownian motion defined by

$$\displaystyle{W_{t} =\int _{ 0}^{t} \frac{\,\overline{\!Z_{s}\!\!}\,\,} {\vert Z_{s}\vert }\,\mathrm{d}Z_{s}\;.}$$

The natural filtration \(\mathcal{F}^{W}\) of W is the filtration generated by Z up to an arbitrary rotation. We show that given any two different matrices Q 1 and Q 2 in O 2(R), there exists an \(\,\mathcal{F}^{Z}\)-previsible process H taking values in {Q 1, Q 2} such that the Brownian motion ∫ H ⋅ dW generates the whole filtration \(\,\mathcal{F}^{Z}\). As a consequence, for all a and b in R such that \(\,a^{2} + b^{2} = 1\), the Brownian motion a(W) + b(W) is complementable in \(\,\mathcal{F}^{Z}\).

Keywords

  • Brownian filtrations
  • Complementability
  • Planar Brownian motion
  • Skew-product decomposition

AMS classification (2010)

  • 60J65
  • 60H20

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Notes

  1. 1.

    A filtration \(\mathcal{F}\) is said to be immersed in a filtration \(\mathcal{G}\) when every \(\mathcal{F}\)-martingale is a \(\mathcal{G}\)-martingale.

  2. 2.

    Given four filtrations \(\mathcal{F}\), \(\mathcal{G}\), \(\mathcal{F}^{{\prime}}\) and \(\mathcal{G}^{{\prime}}\) with \(\mathcal{F}\) immersed in \(\mathcal{G}\) and \(\mathcal{F}^{{\prime}}\) immersed in \(\mathcal{G}^{{\prime}}\), the immersion of \(\,\mathcal{F}^{{\prime}}\) in \(\,\mathcal{G}^{{\prime}}\) is isomorphic to the immersion of \(\,\mathcal{F}\) in  \(\,\mathcal{G}\) if \(\mathcal{G}\) and \(\mathcal{G}^{{\prime}}\) are in correspondence by some isomorphism which maps \(\mathcal{F}\) onto \(\mathcal{F}^{{\prime}}\).

  3. 3.

    A k-dimensional \(\mathcal{Z}\)-BM B is called maximal if no other k-dimensional \(\mathcal{Z}\)-BM generates a strictly bigger filtration than B.

References

  1. J. Brossard, M. Émery, C. Leuridan, Maximal Brownian motions. Ann. de l’Institut Henri Poincaré Probab. Stat. 45(3), 876–886 (2009)

    CrossRef  MATH  Google Scholar 

  2. J. Brossard, C. Leuridan, Filtrations browniennes et compléments indépendants. Séminaire de Probabilités XLI. Lect. Notes Math. 1934, 265–278 (2008). Springer

    Google Scholar 

  3. L. Dubins, J. Feldman, M. Smorodinsky, B. Tsirelson, Decreasing sequences of σ-fields and a measure change for Brownian motion. Ann. Probab. 24(2), 882–904 (1996)

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. M. Émery, On certain almost Brownian filtrations. Ann. de l’Institut Henri Poincaré Probab. Stat. 41(3), 285–305 (2005)

    CrossRef  MATH  Google Scholar 

  5. M. Malric, Filtrations quotients de la filtration brownienne. Séminaire de Probabilités XXXV. Lect. Notes Math. 1755, 260–264 (2001). Springer

    Google Scholar 

  6. D. Stroock, M. Yor, On extremal solutions of martingale problems. Ann. Scientifiques de l’École Normale Supérieure 13(1), 95–164 (1980)

    MathSciNet  MATH  Google Scholar 

  7. D.W. Stroock, M. Yor, Some remarkable martingales. Séminaire de Probabilités XV. Lect. Notes Math. 850, 590–603 (1981). Springer

    Google Scholar 

  8. B. Tsirelson, Triple points: from non-Brownian filtrations to harmonic measures. Geomet. Funct. Anal. 7(6), 1096–1142 (1997)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author gratefully acknowledges the support of the ANR programme ProbaGeo.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Christophe Leuridan .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Brossard, J., Émery, M., Leuridan, C. (2014). Skew-Product Decomposition of Planar Brownian Motion and Complementability. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics(), vol 2123. Springer, Cham. https://doi.org/10.1007/978-3-319-11970-0_15

Download citation