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Pathwise identity for bounded predictable counting-process integrals

Statement

Under Simple counting-process intensity and compensator contract, for every nonnegative predictable $G$ , $E \int_{(0, T]} G_{t} d N_{t} = E \int_{0}^{T} G_{t} d A_{t} .$ For every bounded real predictable $H$ , the L²-completed stochastic integral equals the pathwise finite-variation expression almost surely: $\int_{0}^{T} H_{t} d M_{t} = \sum_{0 < t \leq T} H_{t} Δ N_{t} - \int_{0}^{T} H_{t} d A_{t}, M = N - A .$ The identity holds at every fixed terminal time and coordinatewise for vector or matrix integrands. No left continuity of $H$ is assumed.

Facts & Assumptions

Given: The finite-horizon square-integrable counting-process contract, $N_{0} = A_{0} = 0$ , continuous predictable $A$ , and $E A_{T} < \infty$ ; $G$ is nonnegative predictable and $H$ is bounded predictable.

[F1]

Simple counting-process intensity and compensator contract: $M = N - A$ is a square-integrable martingale, $N$ has finitely many unit jumps, and $A$ is continuous with finite expectation.

[F2]

Progressively measurable and predictable processes: time-zero sets and rectangles $(s, t] \times B$ , $B \in F_{s}$ , generate the predictable sigma-algebra and, with the whole space, form a pi-system.

[F3]

Dynkin's pi-lambda theorem: two finite measures agreeing on a generating pi-system and the whole space agree on its generated sigma-algebra.

[F4]

Monotone convergence for the integral: nonnegative increasing integrals converge to the integral of the limit.

[F5]

Predictable step processes are dense for the compensator L² norm: bounded predictable steps are dense for $E \int H^{2} d A$ .

[F6]

Elementary predictable integral against a counting-process martingale: a step integral is the finite sum of measurable coefficients times increments of $M$ .

[F7]

L² extension of the predictable counting-process integral: step approximation in compensator L² gives the unique completed integral and its isometry.

[F8]

Cauchy-Schwarz inequality for $L^{2}$ : on every measure space, $\int ∣ f g ∣ d μ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ for square-integrable $f, g$ .

[A1]

AC is inherited through the martingale and density interfaces. The Axiom of Choice

Proof

1.1F1F2F3F4givenconstruct

The martingale property gives $E N_{T} = E A_{T} < \infty$ . For a predictable set $E$ , define $μ_{N} (E) = E \int 1_{E} d N$ and $μ_{A} (E) = E \int 1_{E} d A$ . Measurability of the section integrals holds first for the generators in [F2] (their integrals are $1_{B} (N_{t} - N_{s})$ and $1_{B} (A_{t} - A_{s})$ ), then for all predictable sets: the sets with measurable section integrals form a lambda-system, using subtraction from finite terminal masses and increasing partial sums for disjoint unions. Apply [F3] to this generating pi-system. Pathwise countable additivity and [F4] now make $μ_{N}, μ_{A}$ finite measures. For an interval generator, $E [1_{B} (M_{t} - M_{s})] = 0$ by the martingale property, so their masses agree; time-zero masses are both zero and whole-space masses agree. [F3] gives $μ_{N} = μ_{A}$ . Simple functions and [F4] yield the compensation identity for every nonnegative predictable $G$ .

2.1A1F5F6F7F8step 1.1choosealgebra∎

By [F5] choose bounded predictable steps $H_{m}$ with $E \int ∣ H_{m} - H ∣^{2} d A \to 0$ . Step 1.1 and [F8] give $E \int ∣ H_{m} - H ∣ d (N + A) = 2 E \int ∣ H_{m} - H ∣ d A \leq 2 (E A_{T})^{1 / 2} {(E \int ∣ H_{m} - H ∣^{2} d A)}^{1 / 2} ⟶ 0.$ For steps, [F6] identifies the integral with the difference of its pathwise $N$ and $A$ integrals. The displayed bound makes those differences converge in L¹ to the stated pathwise expression. By [F7] the same step integrals converge in L², hence L¹ on the probability space, to the completed stochastic integral. Uniqueness of L¹ limits proves equality almost surely. Apply this argument separately to each fixed terminal time and each of the finitely many coordinates.

Source notes

Bass uses a predictable monotone-class argument for finite-variation jump integrals and defines stochastic integrals by step approximation in L². The local proof above supplies the compensation measure and the L¹ identification needed for arbitrary bounded predictable Cox covariates; it does not import a semimartingale existence theorem.

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Sources

Richard F. Bass, SDEs with Jumps, Sections 2.4–2.5, PDF pp. 5–6 (standard reference, not scraped)