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Cross brackets vanish without simultaneous jumps

Statement

Let $N^{1}, \dots, N^{k}$ be simple unit-jump counting processes on one common filtration, each satisfying the finite-horizon contract of Simple counting-process intensity and compensator contract with continuous compensator $A^{i}$ and martingale $M^{i} = N^{i} - A^{i}$ . If almost surely no two distinct processes jump at the same time, then for $i \neq j$ , $[M^{i}, M^{j}]_{t} = 0 and ⟨ M^{i}, M^{j} ⟩_{t} = 0 (0 \leq t \leq T) .$ No independence assumption is required.

Facts & Assumptions

Given: Finitely many unit-jump counting processes on a common filtration, continuous compensators, the square-integrable martingale contracts, and pairwise absence of simultaneous jumps almost surely.

[F1]

Each $A^{i}$ is continuous and each $M^{i} = N^{i} - A^{i}$ is a martingale. (Simple counting-process intensity and compensator contract)

[F2]

Optional cross brackets are jump-product sums, and predictable cross brackets are continuous predictable finite-variation compensators. (Optional and predictable brackets for counting-process martingales)

[F3]

Continuous predictable brackets in this class are unique. (Uniqueness of continuous predictable brackets)

[A1]

AC is inherited through the martingale and conditional-expectation interface. (The Axiom of Choice)

Proof

1.1F1F2given

For every $s$ , continuity of each $A^{i}$ gives $Δ M_{s}^{i} = Δ N_{s}^{i}$ . The no-simultaneous-jump hypothesis therefore gives $Δ M_{s}^{i} Δ M_{s}^{j} = 0$ for every $s$ and every $i \neq j$ , outside one probability-zero event. By the optional cross-variation definition in [F2], $[M^{i}, M^{j}]_{t} = \sum_{s \leq t} Δ M_{s}^{i} Δ M_{s}^{j} = 0$ for all $t \leq T$ .

2.1A1F2F3step 1.1∎

The zero process is continuous, predictable, has finite variation, and satisfies $[M^{i}, M^{j}] - 0 = 0$ , which is a martingale. Thus it is a continuous predictable cross-bracket candidate. Uniqueness [F3] implies $⟨ M^{i}, M^{j} ⟩ = 0$ . The proof uses the common filtration for the component martingale contracts and uses no independence.

Source notes

Tian states zero cross brackets under a no-simultaneous-jump hypothesis; Andersen–Borgan–Gill–Keiding impose the same condition for multivariate counting processes. The argument here proves both zero bracket identities directly from the jump-sum and predictable-compensator definitions.

Depends on

Used by

Compensators, brackets, and jump-integral isometry for reaction counts Theorem

Dependency tree · two levels

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Sources

Lu Tian, STAT331 Unit 10: Combining Martingales, Stochastic Integrals, and Applications, PDF p. 4 (standard reference, not scraped)
P. K. Andersen, Ø. Borgan, R. D. Gill, N. Keiding, Censoring, Truncation and Filtering in Statistical Models Based on Counting Processes, §2, PDF pp. 3–4 (standard reference, not scraped)