Generalization of Doob Dynkin for Stochastic processes

Question

Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is there any simple representation of the process $Y$ in terms of $X$. This will be a generalization of Doob Dynkin Lemma

Answer 1

I don't think there is a non-trivial result in the general setting, but for some special cases of $X_t$ and $Y_t$ it is possible to represent $Y_t$ as an Ito integral w.r.t. $X_t$, see for example here.

Answer 2

Apart from the trivial consequence of Doob-Dynkin's lemma that for each $t$ there exists some measurable function $u_t$ such that $Y_t=u_t(X_s;s\leqslant t)$... Since every $(Y_t)$ thus defined satisfies the hypothesis, I doubt one can go much further.