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 $A \notin \mathcal{F}_{\tau}$.
  • 1
    $\begingroup$ 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 $t$? $\endgroup$ – user1 Jun 29 '18 at 5:39
  • 1
    $\begingroup$ @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...) $\endgroup$ – saz Jun 29 '18 at 6:42
  • $\begingroup$ it is not, it is just a little more elaborate imo, but thats just a matter of taste. Thank for verifying. $\endgroup$ – user1 Jun 29 '18 at 6:50
  • $\begingroup$ Considering the second paragraph. Does the stopping time really have had to occured? isnt it sufficent to know weather if has occured or not? $\endgroup$ – user1 Oct 28 '18 at 7:33
  • 2
    $\begingroup$ @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 $\omega \in A^c$. $\endgroup$ – saz Oct 28 '18 at 7:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.