CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Optional stopping fails for an unbounded simple-random-walk hitting time

Statement

Assume AC. For simple symmetric random walk $S_{0} = 0$ and $τ = \inf {n \geq 0 : S_{n} = 1}$ , define $S_{τ} = 0$ on ${τ = \infty}$ . Then $τ$ is almost surely finite, but $S_{τ} = 1$ almost surely and $E S_{τ} = 1 \neq 0 = E S_{0} .$

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Gambler's ruin hitting probability from optional stopping computes finite-interval hitting probabilities.

[F2]

Optional stopping requires a passage-to-the-limit hypothesis identifies the invalid limit passage.

[F3]

The Axiom of Choice is inherited from F1 and the martingale interface.

Counterexample

1.1F1

For $a \geq 1$ , let $E_{a}$ be the event that the walk hits $1$ before $- a$ . Translating the symmetric ruin interval to ${0, a + 1}$ with starting point $a$ , F1 gives $P (E_{a}) = \frac{a}{a + 1} .$ The $E_{a}$ increase, and their union is ${τ < \infty}$ : a path that reaches $1$ has a finite minimum before that time and therefore belongs to some $E_{a}$ . Continuity from below gives $P (τ < \infty) = 1$ .

2.1F2F3step 1.1∎

By definition, $S_{τ} = 1$ on this probability-one event, whereas $S_{0} = 0$ . Thus their expectations differ. This explicitly shows that almost-sure finiteness alone cannot justify passing from $τ \land n$ to $τ$ in bounded optional sampling, as F2 warns. AC has exactly the inherited role in F3.

Depends on

Used by

Almost-surely finite stopping does not imply integrable stopping Counterexample

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

van der Vaart, Martingales, Diffusions and Financial Mathematics, warning after Theorem 2.42, p. 22 (standard reference, not scraped)