Let $\xi_t$, $t \in \mathbb{R}$ be a stochastic process, and $\mathcal{F}_{=t}$, $\mathcal{F}_{\geq t}$, $\ldots$ are $\sigma$-algebras induced by it. I want to find out whether the two following statements are equivalent:
- $\forall t \in \mathbb{R}$, $A \in \mathcal{F}_{\leq t}$, $B \in \mathcal{F}_{\geq t}$ a.s. $$ \mathbb{P}\left(AB\mid \mathcal{F}_{=t}\right) = \mathbb{P}\left(A\mid\mathcal{F}_{=t}\right)\mathbb{P}\left(B\mid\mathcal{F}_{=t}\right) $$
- $\forall t \in \mathbb{R}$, $A \in \mathcal{F}_{\leq t}$ a.s. $$ \mathbb{P}\left(B\mid\mathcal{F}_{\geq t}\right) = \mathbb{P}\left(A \mid\mathcal{F}_{=t}\right) $$ Looking at the first statement, I assume that it means that given a fixed ''present'', any event in the "past" and any event in the "future" are conditionally independent on each other. As far as I understand, it is actually a Markov property, so $\xi_t$ is a Markov process. However, I can't interpret the second statement, so I have no idea how to connect them and looking for help.