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 $W_ t = \int_0^\cdot \frac{Z_s}{|Z_s|} dZ_s$. 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 $O_2(\mathbb{R})$, there exists an $\mathcal{F}_Z$-previsible process $H$ taking values in $\{Q_1,Q_2\}$ such that the Brownian motion $\int_0^\cdot H \cdot dW$ generates the whole filtration $\mathcal{F}_Z$. As a consequence, for all $a$ and $b$ in $\mathbb{R}$ such that $a^2 + b^2 = 1$, the Brownian motion $a \mathrm{Re}(W) + b \mathrm{Im}(W)$ is complementable in $\mathcal{F}_Z$.
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Jean Brossard, Michel Emery, Christophe Leuridan. Skew-Product Decomposition of Planar Brownian Motion and Complementability. Séminaire de Probabilités XLVI. Lecture Notes in Mathematics, vol 2123. Springer, pp.377-394, 2014, ⟨10.1007/978-3-319-11970-0_15⟩. ⟨hal-02272534⟩

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