# Definition of $\sigma$-algebra $\mathcal{F}_\tau$ with $\tau$ a stopping time

If $\tau$ is a stopping time and $(\mathcal{F_t})_{t\in I}\subset \mathcal{F}$ is a filtration, then the $\sigma$-algebra of the $\tau$-past is defined as

$$\mathcal{F}_\tau := \{A\in\mathcal{F} : A\cap \{\tau \leq t\} \in \mathcal{F}_t \text{ for any } t \in I\}\, .$$

What is the reasoning behind this definition? How can one characterize the sets which are in $\mathcal{F}_\tau$? Since $A\cap \{\tau \leq t\} \in \mathcal{F}_t$ but not necessarily $A\subseteq \{\tau \leq t\}$, might it not be the case that we can't decide if $A$ happened at time $\tau$? And can one understand $\mathcal{F}_\tau$ similar to $\mathcal{F}_t$ but with the index (somewhat) randomly chosen? If yes, how?

$A \in \mathcal{F}_t$ means: Given the information up to time $t$, we can decide for some fixed $\omega \in \Omega$ whether $\omega \in A$ (or $\omega \in A^c$), i.e. whether the event $A$ happens (or not).

The interpretation for $A \in \mathcal{F}_{\tau}$ is very similar: Given the information up to time $t$ and given that the stopping time $\tau(\omega)$ already occured before time $t$, we want to decide whether $\omega \in A$ (or $\omega \in A^c$).

Example: Let $(B_t)_{t \geq 0}$ be a stochastic process with continuous sample paths and $B_0=0$. Define

$$\tau := \inf\{t>0; B_t \notin (-1,1)\}.$$

• $A:=\{B_{\tau}=1\} \in \mathcal{F}_{\tau}$: Indeed, given the path up to time $t$ and given that the stopping time occured before time $t$, we can easily decide whether $B_{\tau}=1$ or $B_{\tau}=-1$ (since $(B_t)_{t \geq 0}$ has continuous sample paths, these are the only two possibilities).
• $A := \{B_{\tau+1}=0\} \notin \mathcal{F}_{\tau}$: Obviously, this event requires information about the future (compare this with $\{B_{t+1}=0\} \notin \mathcal{F}_t$) and therefore $A \notin \mathcal{F}_{\tau}$.
• How about thinking about it in this way ; Suppose we know that $A$ has happend or not at $\tau$. If we then know by time $t$ that $\tau$ occured we should also be able to see weather $A$ happend or not. I.e those $\omega$ which make $\tau$ happen at $t$ or before and also are in $A$ should form a set which is observable at $t$? – user1 Jun 29 '18 at 5:39
• @user32423 I don't really see how this is much different from what I wrote in my answer (... at least if I understood your comment correctly...) – saz Jun 29 '18 at 6:42
• it is not, it is just a little more elaborate imo, but thats just a matter of taste. Thank for verifying. – user1 Jun 29 '18 at 6:50
• Considering the second paragraph. Does the stopping time really have had to occured? isnt it sufficent to know weather if has occured or not? – user1 Oct 28 '18 at 7:33
• @user1 If the stopping time has not occured up to time $t$, then the information up to time $t$ is not enough to decide whether $\omega \in A$ or $\omega \in A^c$. – saz Oct 28 '18 at 7:38