Séminaire de Probabilités XLVI pp 377-394

Part of the Lecture Notes in Mathematics book series (LNM, volume 2123) | Cite as

Skew-Product Decomposition of Planar Brownian Motion and Complementability

  • Jean Brossard
  • Michel Émery
  • Christophe Leuridan
Chapter

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 Q1 and Q2 in O2(R), there exists an \(\,\mathcal{F}^{Z}\)-previsible process H taking values in {Q1, Q2} 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 

References

  1. 1.
    J. Brossard, M. Émery, C. Leuridan, Maximal Brownian motions. Ann. de l’Institut Henri Poincaré Probab. Stat. 45(3), 876–886 (2009)CrossRefMATHGoogle Scholar
  2. 2.
    J. Brossard, C. Leuridan, Filtrations browniennes et compléments indépendants. Séminaire de Probabilités XLI. Lect. Notes Math. 1934, 265–278 (2008). SpringerGoogle Scholar
  3. 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)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    M. Émery, On certain almost Brownian filtrations. Ann. de l’Institut Henri Poincaré Probab. Stat. 41(3), 285–305 (2005)CrossRefMATHGoogle Scholar
  5. 5.
    M. Malric, Filtrations quotients de la filtration brownienne. Séminaire de Probabilités XXXV. Lect. Notes Math. 1755, 260–264 (2001). SpringerGoogle Scholar
  6. 6.
    D. Stroock, M. Yor, On extremal solutions of martingale problems. Ann. Scientifiques de l’École Normale Supérieure 13(1), 95–164 (1980)MathSciNetMATHGoogle Scholar
  7. 7.
    D.W. Stroock, M. Yor, Some remarkable martingales. Séminaire de Probabilités XV. Lect. Notes Math. 850, 590–603 (1981). SpringerGoogle Scholar
  8. 8.
    B. Tsirelson, Triple points: from non-Brownian filtrations to harmonic measures. Geomet. Funct. Anal. 7(6), 1096–1142 (1997)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jean Brossard
    • 1
  • Michel Émery
    • 2
  • Christophe Leuridan
    • 1
  1. 1.Institut FourierUniversité Joseph Fourier et CNRSSaint-Martin-d’Hères CedexFrance
  2. 2.IRMA, Université Unique de Strasbourg et CNRSStrasbourg CedexFrance

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