Suppose $X_t$ is the solution of the SDE
$$dX=a(X)dt+b_1(X)dW_1+b_2(X)dW_2$$
$Y_t$ is the solution of the following SDE
$$dY=p(Y)dt+q_1(Y)dW_1+q_2(Y)dW_2$$
Here, $W_1$ and $W_2$ are independent Brownian motions.
I am trying to prove that $X$ and $Y$ are independent. I have proven that $dXdY=0$, will this help? If this is not enough, which is what I thought, are there any other conditions I need to check? or some other ways I can try?
I appreciate any suggestions in advance.