Markov property with respect to a filtration Suppose $\{ X_t: t \in \mathbb{R} \}$ is a stochastic process on a probability space $(\Omega, \mathcal{F}, P)$, and it is adapted to a filtration $\{\mathcal{F}_t \}$ on the probability space.


*

*$\{ X_t\}$ is said to have Markov
property with respect to the
filtration $\{\mathcal{F}_t \}$, if 
$\forall t \in \mathbb{R}$ and
$\forall A \in \mathcal{F}_{\geq
    t}$, $$P(A \mid \mathcal{F}_t) = P(A
    \mid X_t) \text{ a.s.}.$$

*$\{ X_t\}$ is said to have Markov
property with respect to its natural
filtration $\{\mathcal{F}_{\leq t}
    \}$, if  $\forall t \in \mathbb{R}$,
$\forall A_1 \in \mathcal{F}_{\geq
    t}$ and $\forall A_2 \in
    \mathcal{F}_{\leq t}$, $$P(A_1 \cap
    A_2 \mid \mathcal{F}_{=t}) = P(A_1
    \mid  \mathcal{F}_{=t}) \, P(A_2 \mid 
    \mathcal{F}_{=t}) \text{ a.s.}.$$
ADDED: $\mathcal{F}_{\leq t}:= \sigma(\{ X_s: s \leq t \})$, $\mathcal{F}_{\geq t}:= \sigma(\{ X_s: s \geq t \})$ and $\mathcal{F}_{= t}:= \sigma( X_t )$.
I was wondering if it is possible to formulate Markov property with respect to the general filtration $\{\mathcal{F}_t \}$, in a way similar to  that with respect to the natural filtration $\{\mathcal{F}_{\leq t}\}$ defined in 2? 
If yes, why is this new definition equivalent to the definition in 1?
Any references?
Thanks in advance!
 A: I've copied this from page 2 of General Theory of Markov Processes by Michael Sharpe,
with some changes in notation. 
Suppose 
$$P(A_1\cap A_2\,|\,  {\cal F}_{=t} )= P(A_1 \,|\,  {\cal F}_{=t} )P( A_2\,|\,  {\cal F}_{=t} )$$
for all $A_1\in {\cal F}_{\leq t}$ and $A_2\in {\cal F}_{\geq t}$.
Using well known properties of conditional expectations, 
\begin{eqnarray*}
P(A_1\cap A_2)
&=&P(P(A_1\cap A_2\ |\  {\cal F}_{=t}))\cr
&=&P\left( P(A_1\  |\  {\cal F}_{=t})\ P(A_2 \ | \ {\cal F}_{=t})  \right)\cr
&=&P(P(A_2\ |\  {\cal F}_{=t}) ; A_1).
\end{eqnarray*}
As $A_1\in {\cal F}_{\leq t}$ was arbitrary, it follows that 
$$P(A_2\  |\  {\cal F}_{\leq t} ) =P(A_2\ | \  {\cal F}_{=t} )$$
for every $A_2\in {\cal F}_{\geq t}$. 
That is, prediction of future behavior of $X$ based on the entire past is
only as valuable as the predictor based on the present value $X_t$ alone.
A: Conditional independence is the relevant concept. Let $\cal F$ be the natural filtration of $(X_t)$ and $\cal G \supset \cal F$ a bigger filtration. Then $(X_t)$ is Markovian with respect to $\cal G$ means that $\cal F_\infty$ is conditionally independent of $\cal G_t$ given $\sigma(X_t)$. You can check that 1 and 2 (the case when $\cal G=\cal F$) are equivalent to this property by using the classical characterizations of conditional independence.
A: If I understand your question correctly the answer is yes and I guess this would suffice as source (page 2 definition 1) he even proves it equivalent to your other definition:
http://books.google.co.uk/books?id=0o0r21x1Ns8C&pg=PA317&dq=moderate+markov+proces&hl=da&sa=X&ei=pnv9T6zCNMOp0QXRk-yPCQ&ved=0CDUQ6AEwAQ#v=onepage&q&f=false
Hope it's to some help.
