Give 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 $\mathcal{F}_t^W$. For the new process defined by the ito integral $$Y_t = \int_0^t X_s dW_s$$ What general properties we need on $X_t \in L^2(\Omega,\{\mathcal{F}_t\}_t, P)$ in order to say $Y_t$ is adapted to the Brownian motion filtration $\mathcal{F}_t^W$?
$X_t$ itself is adapted to $\mathcal{F}_t^W$ is enough, but can we get more general conditions?
For example, is it true if $X_t$ is the solution of the SODE $$dX_t = a(t, X_t) dt + b dW_t?$$