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

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A last exit time need not be a stopping time

Statement

A last-visit time generally is not a stopping time. For two independent fair coin tosses $X_{1}, X_{2}$ and $F_{n} = σ (X_{1}, \dots, X_{n})$ , let $ρ = \max ({k \in {1, 2} : X_{k} = 1} \cup {0}) .$ Then $ρ$ is not a stopping time.

Facts & Assumptions

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

[F1]

Discrete stopping time requires ${ρ \leq 1} \in F_{1}$ .

[F2]

Independent random elements makes the second toss independent of $F_{1} = σ (X_{1})$ .

Counterexample

1.1F1

The last success occurs no later than time $1$ exactly when the second toss fails. Hence ${ρ \leq 1} = {X_{2} = 0} .$ This event has probability $1 / 2$ .

2.1F1F2step 1.1∎

If it belonged to $F_{1}$ , F2 would make it independent of itself, because it is also an event determined by $X_{2}$ . That would give $1 / 2 = P (A) = P (A)^{2} = 1 / 4$ , impossible. Thus ${ρ \leq 1} \notin F_{1}$ , violating F1. The counterexample is finite and choice-free.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, stopping-time discussion in §2.3 (standard reference, not scraped)