Given a Markovian process $(X_t )_{t\geq 0 }$, is the following property accurate?
$$\mathbb E \left[ f(X_{t_1}, X_{t_2},X_{t_3}) \mid \mathcal F ^X_{t_2}\right] = \mathbb E \left[ f(X_{t_1}, X_{t_2},X_{t_3}) \mid X_{t_1},X_{t_2} \right]$$
Where $0\leq t_1 \leq t_2 \leq t_3$, $\mathcal F^X$ is the filtration generated by $X$ and assuming that $f$ has all desired measurability properties.
If it is positive how to show it ?
Edit By Markovian process I mean that for a bounded measurable function $\Phi$ from a $\mathbb R ^d$ in $\mathbb R$ then
$$ \mathbb E \left[ \Phi (X_u), t\leq u \leq s \mid \mathcal {F_t} \right] =\mathbb E \left[ \Phi (X_u), t\leq u \leq s \mid X_t \right]$$
for $ 0\leq t\leq s $.