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4 — Stopping Times

Last week we covered:

  • Random vectors in $\mathbb{R}^d$
  • Poisson convergence

This week: Stopping Times (Chapter 4, Section 1).

1. Stopping Times: Intuition

A stopping time is a random time, but with an information restriction.

You must be able to decide whether “$T = n$” using only the information available at time $n$.

A stopping time is a map: \(T:\Omega \to \mathbb{Z}^+ = \{0,1,2,\ldots\}.\)

2. Filtration

A filtration is a non-decreasing sequence of sigma-algebras: \(\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \cdots \subseteq \mathcal{F}.\)

It represents information growing over time.

Natural Filtration

Given a sequence of random variables $(X_n)$: \(\mathcal{F}_n = \sigma(X_1,\ldots,X_n), \qquad n\ge 1.\)

This is the typical filtration used in probability theory.

3. Definition of Stopping Time

A random time $T$ is a stopping time relative to $(\mathcal{F}_n)$ if: \(\{T = n\} \in \mathcal{F}_n \qquad \forall n\ge0.\)

Equivalent and often more convenient: \(\{T \le n\} \in \mathcal{F}_n.\)

Since \(\{T\le n\} = \bigcup_{k=0}^n \{T=k\}.\)

And \(\{T=n\} = \{T\le n\} \setminus \{T\le n-1\}.\)

4. Examples

Example 1: First time entering a set

Let $A$ be a Borel set and define: \(T = \inf \{k\ge 0 : X_k \in A\}.\)

Then \(\{T=n\} = \{X_0\in A^c,\ldots, X_{n-1}\in A^c, X_n\in A\} \in \mathcal{F}_n.\)

Thus $T$ is a stopping time.

Example 2: Last time in a set (NOT a stopping time)

Define: \(T = \sup \{k\ge 0 : X_k \in A\}.\)

This depends on future values (you need to know whether $X_{n+1}, X_{n+2},\ldots$ ever return to $A$).

Thus $T$ is not a stopping time.

Heuristic:
“Turn left one block before the big intersection” is not a stopping time; it requires knowing the future.
“Turn left at the 4th street” is a stopping time.

Example 3: Constant stopping time

If $T=c$ is a constant, then $T$ is trivially a stopping time, since
\(\{T=c\} = \Omega \in \mathcal{F}_c.\)

5. Sigma-Algebra at a Stopping Time

We define: \(\mathcal{F}_T = \{A \in \mathcal{F} : A\cap \{T=n\} \in \mathcal{F}_n \text{ for all } n\}.\)

Interpretation:
Information available at the random time $T$.

Natural Filtration Case

If $\mathcal{F}_n = \sigma(X_1,\ldots,X_n)$, then: \(\mathcal{F}_T = \sigma\left( X_{n \wedge T} : n\ge 0 \right).\)

This captures the process “stopped at time $T$”.

6. Properties of Stopping Times

If $S$ and $T$ are stopping times relative to $(\mathcal{F}_n)$:

  1. Minimum is a stopping time \(S\wedge T = \min(S,T) \text{ is a stopping time}.\)

    Proof:
    \(\{S\wedge T = n\} = \big( \{S=n\} \cap \{T\ge n\} \big) \cup \big( \{T=n\} \cap \{S\ge n\} \big) \in \mathcal{F}_n.\)

  2. Maximum is a stopping time
    (similar argument).

  3. Order implies filtration inclusion \(S\le T \quad \Rightarrow \quad \mathcal{F}_S \subseteq \mathcal{F}_T.\)

7. Adapted Processes and Stopping Times

A process $(X_n)$ is adapted if: \(X_n \in \mathcal{F}_n, \qquad n\ge1.\)

If $(X_n)$ is adapted and $T$ is a stopping time, then: \(X_T \in \mathcal{F}_T.\)

Proof: \(\{X_T \in B\} = \bigcup_{n=0}^\infty \big( \{X_n \in B\} \cap \{T=n\} \big) \in \mathcal{F}_T.\)

8. Summary

  • Stopping times describe random times determined by present and past information only.
  • Filtrations capture growing information.
  • $\mathcal{F}_T$ represents the information available at the random time $T$.
  • Adapted processes remain measurable when evaluated at stopping times.
  • First hitting times of sets are classic examples.
  • “Last visit times” are not stopping times.

This concludes Lecture 4.

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