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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!

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  • $\begingroup$ Hint: try $x B_2 - y B_1$ with some $x,y$. $\endgroup$
    – zhoraster
    Commented Nov 1, 2017 at 13:42

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