Representations of Itô Processes

Let $(\Omega,\mathscr{F},\mathbf{P})$ be a complete probability space. By an Itô process relative to an increasing family $\{\mathscr{F}_t\}$ of sub-$\sigma$-algebras of $\mathscr{F}$, we mean a process $\xi$ of the form
$$ \xi_t=\xi_0+\int_0^t\alpha_s\,ds+\int_0^t \beta_s\,dW_s $$
where $\alpha,\beta$ are measurable processes well adapted to $\{\mathscr{F}_t\}$, $\displaystyle\int_0^t (|\alpha_s|+\beta_{s}^2)ds<\infty$ $\forall\,t$ a.s., and $W$ is a standard Wiener process with respect to $\mathscr{F}$. We study conditions under which an Itô process $\xi$ relative to $\{\mathscr{F}_t\}$ is also an Itô process relative to a family $\{\mathscr{G}_t\}$ of “simpler” $\sigma$-algebras: $\mathscr{G}_t\subseteq\mathscr{F}_t$ for each $t$.