CounterexampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generated

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An adapted process need not be a martingale

Statement refuted

Assume AC. The assertion that every adapted integrable real process is a martingale is false. A counterexample is $X_{n} = n$ on a one-point probability space.

Facts & Assumptions

Given: The hypotheses and conventions in the statement refuted.

[F1]

A Dirac measure at a specified point is a probability measure. A Dirac set function is a probability measure.

[F2]

Conditional expectation is linear, order preserving and expectation preserving. Basic algebra and order properties of conditional expectation.

[F3]

AC supplies the inherited conditional-expectation existence and any stated choice of versions. The Axiom of Choice.

Counterexample

technique · direct

1.1

Let $Ω = {*}$ with $P = δ_{*}$ and $F_{n} = {\emptyset, Ω}$ for all $n$ . This is a probability space and a filtration. Define $X_{n} (*) = n$ . Each $X_{n}$ is measurable for $F_{n}$ and $E ∣ X_{n} ∣ = n < \infty$ , so the process is adapted and integrable at every time.

2.1

Conditional expectation fixes constants, so $E [X_{n + 1} ∣ F_{n}] = n + 1$ , while $X_{n} = n$ . The two values differ on $Ω$ , which has probability one, at every $n$ , including $0$ . Hence the martingale equality Martingale submartingale and supermartingale fails. AC is inherited from the CE class convention; the singleton calculation itself is explicit.

F2F3step 1.1∎

Depends on

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Sources

Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)