Let $(\Omega,\mathcal{F},\mathbb{P})$ be a finite probability space, $\mathbb{F}=\{\mathcal{F}_{n}\}_{n=1,\ldots,N}$ a filtration, and $X=\{X_{n}\}_{n=1,\ldots,N}$ a discrete-time and -space process adapted to $\mathbb{F}$.
Moreover, let $\mathbb{G}=\{\mathcal{G}_{n}\}_{i=1,\ldots,N}$ be a filtration with $\mathcal{G}_{n}\subset\mathcal{F}_{n}$ and $Y=\{Y_{n}\}_{n=1,\ldots,N}$ a process adapted to $\mathbb{G}$.
I am interested in condtions on $X$ and $\mathbb{G}$ that imply $$\tag{1}\mathbb{E}[\, F(X_{1},\ldots,X_{m},Y_{1},\ldots,Y_{m}) \;|\; \mathcal{F}_{n} \,] \;=\; \mathbb{E}[\, F(X_{1},\ldots,X_{m},Y_{1},\ldots,Y_{m}) \;|\; \mathcal{G}_{n} \,]$$ for all $m\geq n$ and functions $F$.
Example: let $X$ be a Markov chain and $\mathbb{G}$ its natural filtration. Then $Y_{m}$ is some function of $\{X_{n}\}_{n=1,\ldots,m}$ and so is $F(X_{1},\ldots,X_{m},Y_{1},\ldots,Y_{m})$. Equ. (1) above then holds by the Markov property of $X$.
Are there less restrictive conditions? For example:
What about the case where $\mathbb{G}$ is generated by some Markov process (not necessarily $X$) and $X$ is adapted to $\mathbb{G}$. Does (1) still hold?
Let's assume $$\mathbb{E}[\, F(X_{1},\ldots,X_{m}) \;|\; \mathcal{F}_{n} \,] \;=\; \mathbb{E}[\, F(X_{1},\ldots,X_{m}) \;|\; \mathcal{G}_{n} \,]$$ for all $m\geq n$. Does (1) follow immediately?
Thank you very much for comments, hints and references!
I think we need to require that $X$ be adapted to $\mathbb G$. If that's not the case, then $X_n \not \in \mathcal{G}_n$ for some $n$; choose the indicator function $F(X_1, \dots, X_m, Y_1, \dots, Y_m) = X_n$. Note that the left hand side of (1) will be $X_n$, but the right-hand side will be $\mathbb{E}[X_n | \mathcal{G}_n]$, which is different.