I was reading the book "Studies in the theory of random processes" and in an existence proof for SDE's, the author claims that to establish the existence of solutions to the SDE, it is enough to establish the existence of a solution with $B(t)$ replaced by $B'(t)$ where $B'(t)$ has the same finite dimensional distribution as $B(t)$ where $B$ is the standard brownian motion. Why is this true?
$$dX_t=\mu(t,X(t))+ \sigma(t,X_t) d(B(t))$$