Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE. 

Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{F}_t$. In the typical examples for weak solutions one has $\sigma(B_s | s \le t) \subset \sigma(X_s | s \le t)$, since the Brownian motion is constructed in terms of $X$.

Can I, given a weak solution as above, always replace $\mathcal{F}_t$ by $\sigma(X_s | s \le t)$ and still have a weak solution?

*I am asking this specifically since to me it seems that many of the major textbooks (Karatzas&Shreve, Rogers&Williams) assume or just state as trivial that a weak solution solves the Martingale problem on $C([0,\infty))$ w.r.t. to the filtration $\mathcal{B}_t = \sigma(y(s) | s \le t)$. However this filtration corresponds to $\sigma(X_s | s \le t)$ and not to the filtration $\mathcal{F}_t$ supplied by the weak solution.*