LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

First hitting time of an adapted process is a stopping time

Statement

If $X$ is adapted and $B \subseteq R$ is Borel, then $τ = \inf {n \geq 0 : X_{n} \in B}, \inf \emptyset = \infty,$ is a stopping time.

Facts & Assumptions

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

[F1]

Adapted and integrable stochastic process gives ${X_{k} \in B} \in F_{k}$ .

[F2]

Equivalent event tests for a discrete stopping time reduces the claim to the finite-horizon events.

Proof

1.1

For each $n$ , ${τ \leq n} = ⋃_{k = 0}^{n} {X_{k} \in B} .$ Every term is in $F_{k} \subseteq F_{n}$ by F1, and the union is finite.

2.1

Hence the event belongs to $F_{n}$ , so F2 proves that $τ$ is a stopping time. If the path never enters $B$ , it belongs to none of these events and the convention gives exactly $τ = \infty$ .

F2step 1.1∎

Depends on

Used by

Gambler's ruin hitting probability from optional stopping Corollary
First exit time from an interval Example
Gambler's ruin probability for a biased walk Example

Dependency tree · two levels

7 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)