Let $(X_t: t \geq 0)$ be a (real-valued) Markov process, and let $P^x$ denote the law of $X$ started from $x$, with corresponding expectation $E^x$. Is the following identity true?
$$E(Y|X_t) = E^x(Y) \bigg\vert_{x=X_t}$$
(where $E:= E^0$, and $Y$ is a random variable that is $\sigma(X_s)$-measurable for some $s>t$).
Many thanks for your help.
Because $Y$ is $\sigma(X_s)$-measurable, there is a Borel function $g:\Bbb R\to\Bbb R$ such that $Y=g(X_s)$. You then have (by the simple Markov property) $$ E(Y|X_t) = E^x(g(X_{s-t}))\Big|_{x=X_t}. $$