DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Discrete stopping time

Definition

For a filtration $(F_{n})_{n \geq 0}$ , a map $τ : Ω ⟶ N_{0} \cup {\infty}$ is a stopping time if ${τ \leq n} \in F_{n}$ for every $n \geq 0$ . It is bounded if some deterministic $N$ satisfies $τ \leq N$ almost surely, and finite if $P (τ < \infty) = 1$ . These notions are distinct: finite does not mean bounded.

The event condition is literal, not merely modulo null sets; no completion of the filtration is assumed. Measurability as an extended-integer-valued random variable follows from the equivalent event tests below.

Depends on

Used by

A last exit time need not be a stopping time Counterexample
Sigma-algebra at a stopping time Definition
Stopped random variable and stopped process Definition
Equivalent event tests for a discrete stopping time Lemma
Minimum, maximum, and bounded shifts of stopping times Lemma

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, §2.3, pp. 9–11 (standard reference, not scraped)