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 The natural filtration FW 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 FZ-previsible process H taking values in {Q1;Q2} such that the Brownian motion ∫ H ∙ dW generates the whole filtration FZ. As a consequence, for all a and b in R such that a2+b2 =1, the Brownian motion aR(W)+S(W) is complementable in FZ.