Skew-Product Decomposition of Planar Brownian Motion and Complementability

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  ℜ(W) + b  ℑ(W) is complementable in $\,\mathcal{F}^{Z}$.