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

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Closed martingale characterization

Statement

Assume AC. For a martingale $M$ , the following are equivalent:

${M_{n} : n \geq 0}$ is uniformly integrable;
$M_{n}$ converges in $L^{1}$ to some $M_{\infty}$ ;
there is $X \in L^{1}$ with $M_{n} = E [X ∣ F_{n}]$ almost surely for every $n$ .

In this case one may take $X = M_{\infty}$ , and $M_{n} \to M_{\infty}$ almost surely.

Facts & Assumptions

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

[F1]

Uniformly integrable martingale convergence proves almost-sure and $L^{1}$ convergence from uniform integrability.

[F2]

Multistep martingale characterization gives $M_{n} = E [M_{m} ∣ F_{n}]$ for $m \geq n$ .

[F3]

Conditional lp contraction at $p = 1$ makes conditioning an $L^{1}$ contraction.

[F4]

Uniform integrability of conditional expectations of one variable says that the conditional expectations of one fixed $L^{1}$ variable are uniformly integrable.

[F5]

The Axiom of Choice states AC, assumed here because F1--F4 use conditional expectations and, when displayed simultaneously, chosen countable families of representatives.

Proof

1.1F1

Assume (1). F1 supplies $M_{\infty} \in L^{1}$ with convergence both almost surely and in $L^{1}$ , proving (2) and the final convergence assertion.

1.2F2F3

Assume (2) and fix $n$ . For every $m \geq n$ , F2 gives $M_{n} = E [M_{m} ∣ F_{n}]$ . By F3, $∥ E [M_{m} ∣ F_{n}] - E [M_{\infty} ∣ F_{n}] ∥_{1} \leq ∥ M_{m} - M_{\infty} ∥_{1} \to 0.$ Consequently $M_{n} = E [M_{\infty} ∣ F_{n}]$ almost surely. Thus (3) holds with $X = M_{\infty}$ .

1.3F4

Assume (3). F4 applied to the single variable $X$ makes ${E [X ∣ F_{n}] : n \geq 0}$ uniformly integrable. These variables are the $M_{n}$ , so (1) follows.

2.1F5step 1.1step 1.2step 1.3∎

Steps 1.1, 1.2, and 1.3 prove every direction, including the claimed choice of terminal variable. AC is used exactly through F5; no stronger limiting assertion is made for a merely $L^{1}$ -bounded martingale.

Depends on

Used by

Almost-sure martingale convergence need not preserve expectation Counterexample
Levy upward convergence of conditional expectations Theorem
Optional stopping under uniform integrability Theorem

Dependency tree · two levels

21 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, Theorem 2.23 and discussion, pp. 15–16 (standard reference, not scraped)