Given the filtration $\{\cal{F_t}\}_{t\geq 0}$, there are two Brownian Motions $W_t, \hat W_t$, such that the correlation between $W_t$ and $\hat W_t$ is constant $0<\rho<1$. I try to write $$\hat W_t~=~\rho W_t + \sqrt{1-\rho^2} B_t$$
where $$B_t\doteq \frac{\hat{W_t} - \rho W_t}{\sqrt{1-\rho^2}}.$$
Can we prove that in this decomposition, $B_t$ is a Brownain motion that independent of $W_t$?
I can prove the correlation between $W_t$ and $B_t$ is $0$, so can we conclude they are independent? One way is to prove $(W_t,B_t)$ are Gaussian, can we prove this assertion?
Thanks a lot!