Consider an SDE $$dX_t=b(X_t)dt+\sigma(X_t)dW_t \quad X_0=x \in \mathbb R^n.$$ with $b \colon \mathbb R^n \to \mathbb R^n, \sigma \colon \mathbb R^n \to \mathbb R^{n \times q},$ $W$ is an $R^q$ valued Brownian Motion.
A weak solution to this SDE is $(\Omega,\mathcal F,\mathcal F_t,P,X_t,W_t),$ where $(\Omega,\mathcal F,\mathcal F_t,P)$ is a filtered probability space with an $R^q$ valued Brownian Motion $W$ on it and $X_t$ is $\mathcal F_t$-adapted process. Here, clearly, $X_t$ is adapted to the filtration generated by $W_t$.
So, in this case, one can always choose the filtration $\mathcal F_t$ to be the filtration generated by $W_t$. Hence, finding a weak solution to SDE is equivalent to find $(\Omega,\mathcal F,P,X_t,W_t),$ and just take $\mathcal F_t$ to be the filtration generated by $W_t$. Am I correct?