I dont know where to start here and I cant find anything about it online. To me it seems that is should be true that an stochasic differential equation that driven by a Browinan motion generetes the same filatration, any uncertanity coming from the Browinan motion.
For simplicity we can consider a stochasic differential equation of the form
$dX_{t}=a(t,X_{t})dt+b(t,X_{t})dB_{t}$
where $B_{t}$ is the Brownian motion
This is indeed a subtle point. We follow Karatzas-Shreve section 5.3 "Weak solutions".
By definition (2.1) a strong solution $X_{t}$ is adapted with respect to
$$\mathcal{F}_{t}:=\sigma(\sigma(\xi)\vee \mathcal{F}^{B}_{t}\vee \mathcal{N} ),$$
where $\xi$ is the initial condition and $\mathcal{N}$ is the collection of null sets in $\mathcal{F}^{B}_{\infty}$.
In the case of weak solution $(X,W)$, we first fix some filtration $\mathcal{F}_{t}$ and then ask that both $X$ and the driving BM $W$ are adapted to it. That allows for the possibility of not equaling the above augmentation:
$$\mathcal{F}_{t}\supsetneq \sigma(\sigma(\xi)\vee \mathcal{F}^{W}_{t}\vee \mathcal{N} ).$$
A standard example is the SDE
$$X_{t}=\int_{0}^{t}sgn(X_{s})dW_{s}.$$
We will record the details. First we start with a Brownian motion $(X_{t},\mathcal{F}_{t}^{X})$. Then we let
$$W_{t}:=\int_{0}^{t}sgn(X_{s})dX_{s}.$$
This implies that $W_{t}$ is BM too with some filtration $\mathcal{F}_{t}^{W}\subseteq \mathcal{F}_{t}^{X}$. We will show that this is sharp
$$\mathcal{F}_{t}^{W}\subsetneq \mathcal{F}_{t}^{X}$$
by getting a contradiction. By Tanaka's formula
$$W_{t}:=\int_{0}^{t}sgn(X_{s})dX_{s}=|X_{t}|-2L_{t}^{X}(0)$$
$$=|X_{t}|-\lim_{\epsilon\to 0}\frac{1}{2\epsilon}meas\{0\leq s\leq t: |X_{s}|\leq \epsilon\},$$
and so
$$\mathcal{F}_{t}^{W}\subseteq \mathcal{F}_{t}^{|X|}.$$
So then we would get
$$\mathcal{F}_{t}^{X}\subseteq \mathcal{F}_{t}^{W}\subseteq\mathcal{F}_{t}^{|X|},$$
which is a contradiction because the absolute value removes the information about the sign.