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Validity under one null law does not give a composite-null test supermartingale

Statement refuted

The claim “a nonnegative process that is a supermartingale under one member of a composite null family is a test supermartingale for the whole family” is false.

Facts & Assumptions

Given: The two Bernoulli laws, filtration, process, and composite null family in the Counterexample. Assume AC for the test-supermartingale convention.

[F1]

Nonnegative test supermartingales: a test supermartingale must satisfy the supermartingale and initial-expectation conditions under every null law.

[A1]

The Axiom of Choice: AC is inherited by the conditional-expectation convention in F1.

Counterexample

Let $Ω = {0, 1}$ , let $P_{1 / 4} (X = 1) = 1 / 4$ and $P_{3 / 4} (X = 1) = 3 / 4$ , and take the composite null family $P_{0} = {P_{1 / 4}, P_{3 / 4}}$ . Let $F_{0} = {\emptyset, Ω}$ and $F_{t} = 2^{Ω}$ for $t \geq 1$ . Set $M_{0} = 1$ and $M_{t} = 2 X$ for $t \geq 1$ .

1.1givenF1A1algebra

Under $P_{1 / 4}$ , $F_{0}$ is trivial, so $E_{P_{1 / 4}} [M_{1} ∣ F_{0}] = E_{P_{1 / 4}} [2 X] = 1 / 2 \leq M_{0}$ . At all later times $M_{t}$ is constant in time. The process is nonnegative, adapted, integrable, and starts at one, so it is a test supermartingale for the singleton null ${P_{1 / 4}}$ .

1.2givenalgebra

Under $P_{3 / 4}$ , $E_{P_{3 / 4}} [M_{1} ∣ F_{0}] = E_{P_{3 / 4}} [2 X] = 3 / 2 > M_{0} = 1$ , so the supermartingale inequality fails already at time zero.

2.1F1step 1.2∎

By F1, a test supermartingale for $P_{0}$ must satisfy the supermartingale condition under both laws. Step 1.2 shows failure under $P_{3 / 4}$ , so the proposed process is not a test supermartingale for the composite null.

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