Assume that $S_t$ is an Itô-process with dynamics
$dS_t=\mu(S_t,t)dt+\sigma(S_t,t)dB_t,$
where $\mu,\sigma$ are jointly Borel measurable and are such that $|\mu(x,t)|+|\sigma(x,t)|\le K(1+|x|)$, $|\mu(x,t)-\mu(y,t)|+|\sigma(x,t)-\sigma(y,t)|\le D|x-y|$.
Let $f$ be a borel function such that $E[|f(S_T)]]<\infty$.
Let $t \le T$.
Define $S^{x,t}$, for $s\ge t$ as
$S^{x,t}_s=x+\int_t^s\mu(S^{x,t}_{t'},t')dt'+\int_t^s \sigma(S^{x,t}_{t'},t')dB_{t'}$.
Then I want to show that
$$E[f(S_T)|\mathcal{F}_t]=E[f(S^{x,t}_T)]_{x=S_t}.$$
I think I can assume that $S_t$ is a Markov process?, so it suffices to show that
$$E[f(S_T)|\sigma(S_t)]=E[f(S^{x,t}_T)]_{x=S_t}?$$
There are two things I need to check then. If we define $g(x)=E[f(S^{x,t}_T)]$, then we need that
$$E(|g(S_t)|)<\infty?$$
Do you see how to do that?
The other thing we need to check is that for a borel set $B \in \mathcal{B}(\mathbb{R})$ we have
$$E[f(S_T(\omega))\cdot 1_{S_t^{-1}(B)}(\omega)]=E[g(S_t(\omega))\cdot 1_{S_t^{-1}(B)}(\omega)].$$
Do you see how to prove this condition?
Update:
I see that we may not need to show that $E(|g(S_t)|)<\infty$. If we are able to prove the last statement for bounded $f$ first. Then we can use $f=f^+-f^-$ and further look at $f_m=\min(f^+,m)$ and then $E[f^+(S_t(\omega))\cdot 1_{S_t^{-1}(B)}(\omega)]=E[g(S_t(\omega))\cdot 1_{S_t^{-1}(B)}(\omega)]$ with the monotone convergence theorem(here $g$ is defined using $f^+$, and we define $g_m$ likewise.)
So it remains to prove that $$E[f(S_T(\omega))\cdot 1_{S_t^{-1}(B)}(\omega)]=E[g(S_t(\omega))\cdot 1_{S_t^{-1}(B)}(\omega)]$$ for positive and bounded $f$? Do you see how to do that?
Update 2:
We may also need to assume that there exist a positive process $R_t$ such that $S_t\cdot R_t$ is a martingale. Here $R_t$ is the discounting from finance.
