[Math] Markov property with respect to a filtration

probability theorystochastic-processes

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.

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

  2. $\{ X_t\}$ 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!

Best Answer

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.

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