Verifying the interpretation of stopping times and stopping time $\sigma$-algebras

I have been thinking about the intuition of stopping times and stopping time $\sigma$-algebras. While I feel more or less comfortable with the former notion, I would like to get more insight in the latter. Having read different intuitive explanations, I tried to come up with the following interpretations which I would like to be verified.

Let $\mathbb{F} = {(\mathcal{F}_n)}_{n \in \mathbb{N}_0}$ be a filtration in $(\Omega, \mathcal{F})$. A random variable $\tau : \Omega \rightarrow \mathbb{N}_0 \cup \{ \infty \}$ is called a stopping time if $\{ \tau \leq n \} \in \mathcal{F}_n$ for all $n \in \mathbb{N}_0$.

It can easily be shown that $\{ \tau \leq n \} \in \mathcal{F}_n \quad \forall n \in \mathbb{N}_0 \iff \{ \tau = n \} \in \mathcal{F}_n \quad \forall n \in \mathbb{N}_0$.

Interpretation I have come up with:

The relation $\{ \tau \leq n \}\in \mathcal{F}_n$ means that all the elementary events $\omega$ in the case of which I stop before time $n$ or at $n$ comprise an event in $\mathcal{F}_n$. This means, in particular, that at time $n$, having observed the current event I am at, I know precisely whether I have or whether I have not stopped before time $n$ or at $n$.

For example, suppose that at time $n$ I am on some event $A \in \mathcal{F}_n$. Two mutually exclusive cases are possible:

1. $A \cap \{ \tau \leq n \} = \emptyset\in \mathcal{F}_t$, so the decision to stop before time $t$ or at $t$ has not been made.
2. $A \cap \{ \tau \leq n \} \neq \emptyset$. Additionally, $A \cap \{ \tau \leq n \} \in \mathcal{F}_n$ so $A \cap \{ \tau \leq n \}$ is also an event which I can distinguish at time $n$. So depending on whether I am on $A \cap \{ \tau \leq n \}$ or on $(A \setminus A \cap \{ \tau \leq n \})$, I can tell whether I have or whether I have not stopped before time $n$ or at $n$.

A similar interpretation and example can be given for the relation $\{ \tau = n \} \in \mathcal{F}_n$.

Now consider the stopping time $\sigma$-algebra: $$ \mathcal{F}_{\tau} := \{ A \in \mathcal{F}: A \cap \{ \tau \leq n\} \in \mathcal{F}_n \quad \forall n \in \mathbb{N}_0\} $$ It can indeed be verified that the above family is a $\sigma$-algebra and that $$A \in \mathcal{F}_{\tau} \iff A \cap \{ \tau = n\} \in \mathcal{F}_n \quad \forall n \in \mathbb{N}_0.$$

In literature, usually, $\mathcal{F}_{\tau}$ is described as the $\sigma$-algebra of the events observed up to the stopping the $\tau$, in analogy to $\mathcal{F}_n$, which represents the events observable up to time n.

Interpretation I have come up with:

1. Suppose that at some arbitrary but fixed time $n$ the event $\{ \tau \leq n \} \in \mathcal{F}_{n}$ has occurred which means that there has been a decision to stop before time $n$ or at $n$. Then for every $A \in \mathcal{F}_{\tau}$ I can tell whether $A$ has occurred or not depending on whether I am at the event $A \cap \{ \tau \leq n\} \in \mathcal{F}_{n}$ or not. So the events in $\mathcal{F}_{\tau}$ are those for which I can tell whether they have occurred or not provided that the event $\{\tau \leq n \}$ for some $n \in \mathbb{N}_0$ has occurred (i.e. there has been a decision to stop).
2. Conversely, take any $A \in \mathcal{F}_{\tau}$ and arbitrary but fixed time $n$. Further assume that event $A$ has occured at time t, in the sense that some $B \in \mathcal{F}_n$ has occured with $B \subset A$. So I can tell whether I have or whether I have not stopped before time $n$ or at $n$ depending on whether the event $B \cap (A \cap \{ \tau \leq n \}) \in \mathcal{F}_n$ has occurred or not. (However I think this second point is irrelevant since at every time $n$ I know whether the event $\{\tau \leq n\} \in \mathcal{F}_n$ has occurred or not.)

I feel more or less assured regarding the first interpretation for the stopping time however I am unsure of the interpretation for the $\sigma$-algebra, namely, whether it actually corresponds to the description stated earlier:

$\mathcal{F}_{\tau}$ is the $\sigma$-algebra of the events observed up to the stopping time $\tau$.

So here are my questions:

1. Do you agree with the two interpretations?
2. Can you add something to make them better (especially to that of the $\sigma$-algebra)?
3. Can you come up with different interpretations?

Solution 1:

If $(\mathcal{F}_n)_{n \in \mathbb{N}}$ is the canonical filtration of a stochastic process $(X_n)_{n \in \mathbb{N}}$, then $\mathcal{F}_n$ contains all the information about the process up to time $n$. After observing realizations $X_1(\omega),\ldots,X_n(\omega)$ of the stochastic process, we can decide whether an event $A_n \in \mathcal{F}_n$ has happened, i.e. whether

$$\omega \in A_n \qquad \text{or} \qquad \omega \notin A_n.$$

Since $\{\tau \leq n\} \in \mathcal{F}_n$ this means, in particular, that we can decide whether the stopping has occurred up to time $n$ given the observations $X_1(\omega),\ldots,X_n(\omega)$.

This intuition can be made precise:

Let $(\mathcal{F}_n)_{n \in \mathbb{N}}$ be the canonical filtration of a stochastic process $(X_n)_{n \in \mathbb{N}}$, and let $\tau: \Omega \to \mathbb{N} \cup \{\infty\}$. Then the following statements are equivalent:

1. $\tau$ is a stopping time
2. If $\omega,\omega' \in \Omega$ are such that $\tau(\omega) \leq k$ and $X_j(\omega) =X_j(\omega')$ for all $j=1,\ldots,k$, then $\tau(\omega') \leq k$.

Summary: $\tau$ is a stopping time if the decision to stop before or at time $n$ (i.e. $\tau(\omega) \leq n$) depends only on $X_1(\omega),\ldots,X_n(\omega)$.

Let's turn to $\mathcal{F}_{\tau}$. Fix observations $X_1(\omega),\ldots,X_n(\omega)$. As we have seen in the first part, we then know whether the stopping has occured up to time $n$, i.e. whether

$$\tau(\omega) \leq n.$$

Suppose for the moment being that the stopping has indeed occured before or at time $n$. Then a set $A \in \mathcal{F}$ is in $\mathcal{F}_{\tau}$ if, and only if, we can decide whether $A$ has occurred (given our observations $X_1(\omega),\ldots,X_n(\omega)$).

Example 1: Let $X_n = \sum_{j=1}^n \xi_j$ for random variables $\xi_j$ which are Gaussian with mean $0$ and variance $1$. Define $$\tau := \inf\{n \in \mathbb{N}; X_n < 0\}.$$ Then the set $$\{X_{\tau} \in B\}$$ is in $\mathcal{F}_{\tau}$ for any Borel set $B$. Indeed: Given that we know that the stopping has occured up to time $n$, we can say which values $X_{\tau}(\omega)$ takes, given the observations $X_1(\omega),\ldots,X_n(\omega)$. In contrast, if the stopping has not occured up to time $n$, the observations $X_1(\omega),\ldots,X_n(\omega)$ don't tell us anything about $X_{\tau}(\omega)$.

Example 2: Let $X_n = \sum_{j=1}^n \xi_j$ for random variables $\xi_j$ such that $\mathbb{P}(\xi_j = 1)= 1/4$ and $\mathbb{P}(\xi_j = -1) = 3/4$. If we define

$$\tau := \inf\{n \in \mathbb{N}; X_n = 100\}$$

then

$$A := \{ \exists k \in \mathbb{N}; X_k =95\} \in \mathcal{F}_{\tau};$$

however, for instance,

$$B := \left\{ \max_{k \geq 0} X_k \leq 100 \right\} \notin \mathcal{F}_{\tau}.$$

Solution 2:

I don't think we actually know whether a specific event occurred just because it is in the $\sigma$ algebra. Its complement is in $\mathfrak{F}_n$ too.

As was detailed more thoroughly in the answer of Nate Eldredge on the Intuition behind Conditional Expectation, I'd say that $\{\tau \leq n\} \in \mathfrak{F}_n$ gives us the ability to ask whether we stopped. If we had the realization of our stochastic process nothing would be random. That is just a deterministic function depending on n.

So I think the sentence

"This means, in particular, that at time $n$, having observed the current event I am at, I know precisely whether I have or whether I have not stopped before time $n$ or at $n$."

is wrong since the $\sigma$-Algebra does not contain what you observe but rather what is observable.

Suppose you are blind and pick coloured balls with a unique number from an urn and you have a machine that cannot tell you the number on the ball but only the colour. You cannot see the concrete realization of the process. You can't have full information but the machine could tell you the colour for each pick you make. So the $\sigma$-Algebra would contain something like

$\mathfrak{F}_n = \{ \emptyset, \Omega, R, B \}$

In case we only have two colours Red and Blue. Now suppose your machine got an update and now it can tell you whether the number on the ball is even or not as well. Now we have

$\mathfrak{F}_{n+1} = \sigma (\{R, B, E, \neg E \})$

So I think the $\sigma$-Algebra represents more the capacities of the observer than our knowledge at a certain time. For the stopping time $\tau$ with $\{\tau \leq n\} \in \mathfrak{F}_n$ that would then mean that the observer actually has the capacity to say whether the process is stopped. So if we want to calculate probabilities or conditional expectations we can use this capacity in our analysis.