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

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Optional stopping with integrable time and bounded increments

Statement

Assume AC. Let $M$ be a martingale with $∣ M_{n} - M_{n - 1} ∣ \leq C$ almost surely for every $n \geq 1$ , for deterministic $C < \infty$ . If $τ$ is a stopping time with $E τ < \infty$ , define $M_{τ} = 0$ on the null event ${τ = \infty}$ . Then $M_{τ} \in L^{1}$ and $E M_{τ} = E M_{0}$ .

Facts & Assumptions

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

[F1]

Optional sampling for bounded stopping times gives $E M_{τ \land n} = E M_{0}$ .

[F2]

Expectation of a nonnegative or integrable random variable defines $E τ$ as an extended nonnegative integral; Markov's inequality for random variables bounds $P (τ \geq m)$ by $E τ / m$ for each $m > 0$ .

[F3]

Dominated convergence passes the stopped values in $L^{1}$ .

[F4]

The Axiom of Choice is inherited from the martingale and bounded optional-sampling theorem.

Proof

1.1

For every positive integer $m$ , F2 gives $P (τ = \infty) \leq P (τ \geq m) \leq E τ / m$ . Since $E τ < \infty$ , letting $m \to \infty$ shows $τ < \infty$ almost surely. On that event, $∣ M_{τ \land n} - M_{τ} ∣ \leq C (τ - n)^{+} \leq C τ,$ because the difference telescopes over at most $(τ - n)^{+}$ increments. It tends pointwise to zero and has the integrable dominator $C τ$ .

2.1

F3 gives $M_{τ \land n} \to M_{τ}$ in $L^{1}$ ; in particular $M_{τ}$ is integrable. F1 gives $E M_{τ \land n} = E M_{0}$ for every $n$ , so taking the $L^{1}$ limit proves the equality. Bounded increments and integrability of $τ$ are used exactly in step 1.1. AC has only the role in F4.

F1F3F4step 1.1∎

Depends on

Used by

Almost-surely finite stopping does not imply integrable stopping Counterexample
Optional stopping requires a passage-to-the-limit hypothesis Remark

Dependency tree · two levels

22 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

Durrett, Probability: Theory and Examples, 5th ed., optional stopping criteria in §4.8 (standard reference, not scraped)