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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!

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    Tom - yes. Transition functions are used to define Markov processes.2011-03-10

2 Answers 2

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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\}, $

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    D Sure, by the chain rule for conditional probabilities. But my point is that, since the OP asks for clear and explicit definitions and proofs about a notion he/she is trying to understand in depth, you might want to add to your post an expanded version of the remark you made in the last sentence of your comment.2011-03-11
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For the first one the notation is wrong it should I think $A\cap\{\tau\le t\}\in\mathcal{F}_t$ instead of $A\cap\tau$.

And for the fourth point, look at George Lowther comments below, that fully address the problematic.

Regard

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    I'm guessing, bu$t$ I think the per$s$on who wro$t$e that Wikipedia entry copied a $s$tatement from $s$omewhere which was only dealing wi$t$h Levy processes, and also made some typos.2011-03-10