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!