I have a question regarding that topic: Are Ito Integrals adapted to the Brownian Motion Filtration.
Given a probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,P)$, we could define a 1-dim Brownian motion $W_t$ adapted to $(\mathcal{F}_t)_t$ with its own filtration $\mathcal{F}_t^W$. $X$ an $(\mathcal{F}_t)_t$-adapted process. For the new process defined by the ito integral $Y_t=\int_0^t X_s dW_s$, is it enough for $Y_t$ to be $\mathcal{F}_t^W$-adapted that $X_s\neq 0$, $s\in [0,t]$ a.s.? Or is there also a counterexample?