Séminaire de Probabilités XLVI pp 377-394

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

# 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

60J65 60H20

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© 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