This question arises from this one, and more generally from many textbook asserting that « for any stochastic process having the Markov property, the past and the future are independent given the present. »
Let $X = \{X(t) : t\in \left[0, \infty\right) \}$ be a stochastic process, and suppose that $X$ has the Markov property, i.e. for all $s, t \in \left[0, \infty\right)$ s.t. $s<t$, we have $$ \mathbb{P}(X(t) \in \cdot | F_s) = \mathbb{P}(X(t) \in \cdot | X(s)) .$$
In the other hand, we say of a stochastic process $S = \{S(t) : t\in\left[0,\infty\right)\}$ that the past is independent from the future given the present if the relation $$ \mathbb{E}\left[S(t_1)S(t_3) | S(t_2)\right] = \mathbb{E}\left[S(t_1)|S(t_2)\right] \mathbb{E}\left[S(t_3)|S(t_2)\right] $$
holds as long as $0\leq t_1 < t_2 < t_3$.
The question is: how do you show that the past is independent from the future given the present for $X$? I tried to write $\mathbb{P}(X(t) \in \cdot | X(s))$ as an expectation, but to no avail.