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Uniqueness of continuous predictable brackets

Statement

Let $X$ be an adapted process on $[0, T]$ . Suppose $C, D$ are continuous, predictable finite-variation processes with $C_{0} = D_{0} = 0$ , $E [Var (C)_{T} + Var (D)_{T}] < \infty$ , and both $X - C$ and $X - D$ are martingales. Then $C$ and $D$ are indistinguishable.

Facts & Assumptions

Given: The hypotheses of the Statement and the continuous predictable compensator convention of Optional and predictable brackets for counting-process martingales.

[F1]

A continuous finite-variation martingale with integrable total variation and initial value zero is indistinguishably zero. (A continuous finite-variation martingale is constant)

[F2]

A martingale is an adapted integrable process satisfying the all-pairs conditional-expectation identity. (Continuous-time adapted processes and martingales)

[F3]

Conditional expectation is linear, so differences of martingales are martingales. (Basic algebra and order properties of conditional expectation)

[F4]

Indistinguishability means equality for all times outside one probability-one event. (Process law, modification, and indistinguishability)

[A1]

AC is inherited from the conditional-expectation interface in [F2]. (The Axiom of Choice)

Proof

1.1A1F1F2F3given

The difference $D - C = (X - C) - (X - D)$ is a martingale by [F3]. It is continuous, has finite variation, starts at zero, and $E Var (D - C)_{T} \leq E Var (D)_{T} + E Var (C)_{T} < \infty$ , so it satisfies [F1].

2.1F1F4step 1.1∎

By [F1], $D - C$ is zero outside one probability-one event, which is indistinguishability by [F4]. Hence $C = D$ indistinguishably, proving uniqueness for both scalar and cross-bracket compensators in the stated continuous class.

Source notes

The cited sources formulate predictable brackets through predictable compensators. The local uniqueness proof reduces that convention to the finite-variation martingale lemma, rather than assuming a continuous-time Doob–Meyer uniqueness result not otherwise supplied on this page.

Depends on

Used by

Cited to discharge well-definedness by Optional and predictable brackets for counting-process martingales.

Dependency tree · two levels

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Sources

Richard F. Bass, SDEs with Jumps, Section 2.4, PDF pp. 4–6 (standard reference, not scraped)
Lu Tian, STAT331 Unit 9: Some Key Results for Counting Process Martingales, PDF pp. 9–12 (standard reference, not scraped)