$\sigma$-algebra of $\tau$-past: Definition vs interpretation

Question

Let $(\Omega,\Sigma)$ be a measurable space and $(F_t)_{t\in I}$ a filtration. We postulate that $F_t$ is the information available at time $t$. Lastly, let $\tau$ be a stopping time. According to Wikipedia the $\sigma$-algebra of the $\tau$-past is defined by $$F_\tau:= \{ A \in \Sigma: \forall t :\{ \tau \leq t \} \cap A \in F_t\}$$ and the interpretation is that $F_\tau$ is the information available at time $\tau$. But according to that interpretation, I would rather expect a totally different definition:

Well, actually I have no objection for the case that $\tau=\infty$ for all outcomes. So suppose that $\tau<\infty$ for at least one outcome and $A\in F_\tau$, then I would expect the following equivalence to hold true: $$A\in F_\tau\Leftrightarrow \forall\omega\in\{\tau<\infty\}:A\in F_{\tau(\omega)}$$ For example, we could define $$F_\tau:=\bigcap_{\omega\in\Omega:\tau(\omega)<\infty}F_{\tau(\omega)}$$ such that $F_\tau$ is clearly a $\sigma$-algebra and it ensures that given $A\in F_\tau$ we know $A$ at time $\tau$ no matter the outcome. But I don't see how this fits together with the actual definition.

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$^1$ I.e. $A\in F_t$ if and only if we know $A$ at time $t$ for each outcome (i.e. for all $\omega\in \Omega$), i.e. at time $t$ we can determine whether $A$ happened or not (no matter what the outcome is).

Answer 1

A first interpretation:

Being $A$ a measurable subset of a probability space means that we can know what is the probability that $A$ happens, if $A$ is not measurable then we can't know that, that is, we can't ask something about $A$.

Being $\mathcal{F}_{\tau }$-measurable means that if $\tau $ happens at some moment in $[0,t]$ (for clarity suppose that $t$ is the present, but it could be any time in an abstract sense) then we can ask about the probability that $A$ has already happened also in $[0,t]$, that is, at time $t$ we can compute $$ \Pr [A|\tau \leqslant t]=\frac{\Pr [A\cap \{\tau \leqslant t\}]}{\Pr [\tau \leqslant t]} $$ as far as $\Pr [\tau \leqslant t]>0$. However, in general, we are unable to know this information if $\tau $ has not happened yet at time $t$, that is, in the general case the event $A\cap \{\tau >t\}$ is not $\mathcal{F}_t$-measurable so we are unable to measure what is the probability that $A$ has happened in $[0,t]$ when $\tau $ has not happened yet.

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A second interpretation:

Another way to understand it, probably more clear, is the following: observe that $$ \Pr [A|\mathcal{F}_\tau ]:=\operatorname{E}[\mathbf{1}_{A}|\mathcal{F}_{\tau }]=\mathbf{1}_{A} $$ for any $A\in \mathcal{F}_{\tau }$. Thus, for any given $\omega \in \Omega $ we will know if $\Pr [A|\mathcal{F}_\tau ](\omega )$ is zero or one, that is, we will know "at time $\tau $" if $\omega \in A$ or $\omega \notin A$.

Now we can clarify what means "at time $\tau $" from the definition of $\mathcal{F}_{\tau }$. Notice that $$ \Pr [A\cap \{\tau \leqslant t\}|\mathcal{F}_t]:=\operatorname{E}[\mathbf{1}_{A}\mathbf{1}_{\{\tau \leqslant t\}}|\mathcal{F}_{t}]=\mathbf{1}_{A}\mathbf{1}_{\{\tau \leqslant t\}} $$ What this means? That given $\omega \in \Omega $ then at time $t$ the value of $\mathbf{1}_{A}(\omega )\cdot \mathbf{1}_{\{\tau \leqslant t\}}(\omega )$ is known. This means that when $\tau (\omega)\leqslant t$ (that is, when $\mathbf{1}_{\{\tau \leqslant t\}}(\omega )=1$) then we can know for sure if $\omega \in A$ or $\omega \notin A$. However if $\tau (\omega )>t$ then $\mathbf{1}_{\{\tau \leqslant t\}}(\omega )=0$ so, in general, we can't know if $\omega \in A$ or $\omega \notin A$.

In short: if $\tau $ happens then we can know if any $A\in \mathcal{F}_\tau $ happens or not. However if $\tau $ doesn't happens then in the general case we can't say if some event $A\in \mathcal{F}_{\tau }$ happens or not.

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The difference between the first interpretation and the second is that the abstraction comes in different places. In the first interpretation we don't have a concrete $\omega \in \Omega $, in place of $\omega $ we just have probabilities that an event happens or not, that is, we are talking about what is possible to compute or not. But in the second interpretation there are not, at least directly, probabilities involved if not that we are assuming that some $\omega \in \Omega $ is given and then we ask if given this $\omega $ and some time $t$ we can know if an event happened or not, that is, the abstraction here is in the events, if at time $t$ and given some $\omega $ we can say for sure if $\omega \in A$ or $\omega \notin A$.

Hope now everything is more clear.

Answer 2

I just write my thinking but I don't know if it could solve your problem.

We can transfer the viewpoint to consider a class of samples rather than every sample $\omega\in\Omega$. We partitioned $\Omega$ as $\{\tau=t\},t\in\mathbb{R}_+\cup\{\infty\}$. For an event $A\in\mathscr{F}_\tau$, we want it to satisfy that for $\forall t\in\mathbb{R}_+$ and $\forall\omega\in\{\tau=t\}$, we can judge if $\omega\in A$ or not, i.e. for $\{\tau=t\}$, we have the information about $A\cap\{\tau=t\}$ and $\{\tau=t\}\backslash A$.

Recall the stochastic basis, filtration $\mathbb{F}$ contains all the information available at every time, which are just events we can judge them happened or not at every time $t\in\mathbb{R}_+$. At every time $t\in\mathbb{R}_+$, we always care about stopping happened or not, i.e. $\{\tau\leq t\}$ or $\{\tau >t\}$. When stopping happened, for $\forall\omega\in\{\tau\leq t\}$ we should be able to judge the event $A\in\mathscr{F}_\tau$ happened or not. So we should be able to judge the event $\{\tau\leq t\}\cap A$. At time $t$, all the events we can judge it happened or not are in $\mathscr{F}_t$, so we should have $\{\tau\leq t\}\cap A\in\mathscr{F}_t$ for $\forall t\in\mathbb{R}_+$. That's the requirement for $\sigma$-algebra of $\tau$-past.

Answer 3

I would expect...

I should rather have expected the following equivalence$^1$: $$A\in F_\tau\Leftrightarrow \forall\omega\in\{\tau<\infty\}:\text{At time $\tau(\omega)$ we can determine whether }A\text{ happened or not}$$ The point is that the RHS is not equivalent to $A\in F_{\tau(\omega)}$:

Set $t:=\tau(\omega)$, then $A\in F_t$ means that we know $A$ at time $t$ for all possible outcomes, not only in the case of the outcome $\omega$.

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$^1$ We can at least show that the LHS implies the RHS: Consider an outcome $\omega\in\Omega$ such that $\tau(\omega)<\infty$. Set $t:=\tau(\omega)$, then we know that $\omega\in A$ if and only if $\omega\in A\cap\{\tau\leq t\}$. Since $A\cap\{\tau\leq t\}$ is in $F_t$ (we are assuming that $A\in F_\tau$), we know whether $\omega\in A\cap\{\tau\leq t\}$ and hence we know whether $\omega\in A$.