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I am trying to verify that the object they define as $\mathcal F_\tau$ is indeed a sigma algebra. I am having trouble proving it's closed under complements relative to $\Omega$. I know the meaning of all the terms involved, so those do not need to be explained. (Like sigma algebra, stopping time, etc.) Thanks.

Oops. As pointed out I forgot to include a critical link http://en.wikipedia.org/wiki/Markov_property#Strong_Markov_property

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What do you mean by $\mathcal F_t$, is $t$ a stopping time? –  Mohamad Mar 1 '13 at 2:09

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up vote 2 down vote accepted

Note that there's a typo in Wikipedia's definition of $\mathcal{F}_\tau$. It defines $$\mathcal{F}_\tau = \{ A \in \mathcal{F} : \tau \cap A \in \mathcal{F}_t, t \ge 0 \}$$ but $\tau \cap A$ doesn't make sense. The correct definition should be $$\mathcal{F}_\tau = \{ A \in \mathcal{F} : \{\tau \le t\} \cap A \in \mathcal{F}_t, t \ge 0 \}.$$

For showing that $\mathcal{F}_\tau$ is closed under complements, here's a hint: for any sets $A,B$, we have $A^c \cap B = (A \cap B)^c \cap B$. (Draw a Venn diagram if you like.)

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