Let $X_t$ be an Itō process related to a Brownian motion $B_t$
$$ X_t = \int_0^t a(t) \text{d}B_t $$
where $a(t)$ is a process which is adapted to the natural filtration of $B_t$, say $\mathcal{B}_t$.
Considering the natural filtration $\mathcal{X}_t$ of the process $X_t$, what simple (simple-to-prove) conditions warrant that the filtrations $\mathcal{X}_t$ and $\mathcal{B}_t$ are identical? For instance if $a(t)$ is a non-random positive function, this seems to be true.