if you have two independent Brownian Motions, it is quite easy to create a new Brownian motion with respect to the same filtration so that the new Brownian Motion has a specific correlation to one of the former Brownian Motion. You can do that by orthogonal transformation.
My question is (whether and) how to do the inverse. So let's assume I have a two dimensional BM $(B_1, B_2)$ with $\mathrm{Covar}(B_1,B_2)=\rho$. Can I prove / show or is there a theorem that states that there exists a Brownian Motion $B^\bot$ with respect to the same filtration and which is orthogonal to $B_1$?
Thank you very much for your help!