Let $\xi_t$, $t\in\mathbb{R}$ be a random process, and $\mathcal{F}_{=t}$, $\mathcal{F}_{≥t}$,... be the induced $σ$-algebras. Check equivalence of the following properties:
1) $∀t ∈ \mathbb{R}$, $A ∈ \mathcal{F}_{≤t}$ almost surely $P(A|\mathcal{F}_{≥t}) = P(A|\mathcal{F}_{=t})$
2) $∀t ∈ \mathbb{R}$, $B ∈ \mathcal{F}_{≥t}$ almost surely $P(B|\mathcal{F}_{≤t}) = P(B|\mathcal{F}_{=t})$
What do these two properties mean? I can't see how they give information about the process.
I think it is useful to interpret the $\sigma$-fields in question in terms of information. For example, the second statement tells you that, in order to determine whether an event $B \in \mathcal{F}_{\geq t}$ occurs or not, the $\sigma$-field $\mathcal{F}_{\leq t}$ does not give you more information than what is already contained in $\mathcal{F}_{= t}$. In other words, you will make the same prediction about the occurrence of some event $B$ in the future whether you know the whole past ($\mathcal{F}_{\leq t}$) or just the present ($\mathcal{F}_{=t}$).