[Math] Questions about the concept of strong Markov property

probability theorystochastic-processes

I am trying to understand the concept of strong Markov property quoted from Wikipedia:

Suppose that $X=(X_t:t\geq 0)$ is a
stochastic process on a probability
space
$(\Omega,\mathcal{F},\mathbb{P})$ with
natural filtration
$\{\mathcal{F}\}_{t\geq 0}$. Then $X$
is said to have the strong Markov
property if, for each stopping time
$\tau$, conditioned on the event
$\{\tau < \infty\}$, the process
$X_{\tau + \cdot}$ (which maybe needs
to be defined) is independent from
$\mathcal{F}_{\tau}:=\{A \in \mathcal{F}: \tau \cap A \in \mathcal{F}_t ,\, \ t \geq 0\}$ and
$X_{\tau + t} − X_{\tau}$ has the same
distribution as $X_t$ for each $t \geq 0$.

Here are some questions that make me stuck:

  1. In $\mathcal{F}_{\tau}:=\{A \in
    \mathcal{F}: \tau \cap A \in
    \mathcal{F}_t ,\, \ t \geq 0\} $,
    what does $\tau \cap A $ mean?
    $\tau$ is a stopping time and
    therefore a random variable and $A$
    is a $\mathcal{F}$-measurable
    subset, but what does $\tau \cap A$
    mean?
  2. How is the process $X_{\tau + \cdot}$ defined from the process $X_{\cdot}$ ? Is it the translated
    version of the latter by $\tau$?

  3. How is the conditional independence
    between a process, such as $X_{\tau
    + \cdot}$, and the sigma algebra, such as $\mathcal{F}_{\tau}$, given
    an event, such as $\{\tau <
    \infty\}$, defined?

    Related question, is independence
    between a random variable and a
    sigma algebra defined as
    independence between the sigma
    algebra of the random variable and
    the sigma algebra?

  4. Is "$X_{\tau+ t} − X_{\tau}$ has the
    same distribution as $X_t$ for each
    $t \geq 0$" also conditional on the
    event $\{\tau < \infty\}$?

Thanks and regards!

Best Answer

Here is a less garbled version of the Wikipedia definition. (Use TheBridge's correction for the definition of ${\cal F}_\tau$.) The post-$\tau$ process $X_{\tau+\cdot}$ is defined on the event $\{\tau<\infty\}$ by $$ X_{\tau+t}(\omega) = X_{\tau(\omega)+t}(\omega),\qquad t\ge 0, $$ for $\omega\in\{\tau<\infty\}$. One way to state the strong Markov property is this: The conditional distribution of $X_{\tau+\cdot}$ given ${\cal F}_\tau$ is (a.s.) equal to the conditional distribution of $X_{\tau+\cdot}$ given $\sigma\{X_\tau\}$, on the event $\{\tau<\infty\}$. More precisely, $$ P[ X_{\tau+t}\in B|{\cal F}_\tau] = P[ X_{\tau+t}\in B|X_\tau],\qquad \hbox{almost surely on }\{\tau<\infty\}, $$ for all $t\ge 0$, and all measurable subsets $B$ of the state space of $X$.

This is equivalent to the statement that $X_{\tau+\cdot}$ and ${\cal F}_\tau$ are conditionally independent, given $X_\tau$: $$ P[ F\cap \{X_{\tau+t}\in B\}|X_\tau] = P[ F|X_\tau]\cdot P[X_{\tau+t}\in B|X_\tau],\qquad \hbox{almost surely on }\{\tau<\infty\}, $$

Related Question