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Let $(\mathcal F_n)$ be a filtration and $\tau$ be a stopping time with values in $\mathbb N \cup \{\infty\}$.

Let $\mathcal F_\infty$ be the sigma-algebra generated by $\cup_n \mathcal F_n$. Define $$\begin{align}\mathcal F_\tau &=\{A\in \mathcal F_\infty, \forall n\in \mathbb N,\; A\cap (\tau =n)\in \mathcal F_n \}\\ &=\{A\in \mathcal F_\infty, \forall n\in \mathbb N,\; A\cap (\tau \leq n)\in \mathcal F_n \} \end{align} $$

$\mathcal F_\tau$ is referred to in Klenke as the "$\sigma$-algebra of $\tau$-past", and in Gut as the "pre-$\tau-\sigma$-algebra".

What's the intuition behind $\mathcal F_\tau$ ? Why does it have the names I just mentionned ?

Gabriel Romon
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    $\mathcal F_t$ is all the information up to time $t$. $\mathcal F_\tau$ is all the information up time $\tau$. The only difference is that $\tau$ is random. For example, if you had a random walk, and you wanted to ask "How many times did the random walk hit $-5$ before it first hit $10$?", then letting $\tau$ be the first time the random walk hit 10, $\mathcal F_\tau$ would give you the information to answer that question. – Mike Earnest Sep 25 '17 at 21:01
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    Example: consider $\tau$ the first return to $0$ of your favorite random walk on $\mathbb Z$, $(\mathcal F_n)$ the associated filtration, $A$ the event that the walk hits $42$ before time $\tau$ and $C$ the event that the walk is negative at time $\tau+1$. Then $A$ is in $\mathcal F_\tau$ and $C$ is not. – Did Sep 26 '17 at 15:04
  • @Did The sets you mentioned, how do we know they lie in $\mathcal F_\infty$ since to me, they seem like random sets. – math111 May 07 '20 at 15:33
  • @MikeEarnest see my comment above – math111 May 07 '20 at 15:34
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    Also, see here for a good explanation on the intuition of the stopped $\sigma$-field – roi_saumon Oct 06 '21 at 14:35

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