Let $I \subset \mathbb{R}$ and let $X=\left(X_{t}\right)_{t \in I}$ be a stochastic process. For $t \in I$, define the $\sigma$-algebras that code the past before $t$ and the future beginning with $t$ by
$$
\mathcal{F}_{\leq t}:=\sigma\left(X_{s}: s \in I, s \leq t\right) \quad \text { and } \quad \mathcal{F}_{\geq t}:=\sigma\left(X_{s}: s \in I, s \geq t\right)
$$
Show that $X$ has the Markov property if and only if, for every $t \in I$, the $\sigma$-algebras $\mathcal{F}_{\leq t}$ and $\mathcal{F}_{\geq t}$ are independent given $\sigma\left(X_{t}\right)$ (compare Definition 12.20).
In other words, a process has the (possibly time-inhomogeneous) Markov property if and only if past and future are independent given the present.