Skew-Product Decomposition of Planar Brownian Motion and Complementability

  • Jean Brossard
  • Michel Émery
  • Christophe Leuridan
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)


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 \(\,\mathcal{F}^{Z}\).


Brownian filtrations Complementability Planar Brownian motion Skew-product decomposition 

AMS classification (2010)

60J65 60H20 



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


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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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