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

probability theorystochastic-calculusstochastic-processesstopping-times

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?

Best Answer

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,

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

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