LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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The general Ito integral is well defined

Statement

Assume the Axiom of Choice and the standing hypothesis (H) of Elementary predictable Brownian integrands. Let $H$ be a predictable process on $[0, T]$ with $E \int_{0}^{T} H^{2} d s < \infty$ , and let $(H^{n})$ and $(K^{n})$ be two sequences of bounded elementary predictable integrands converging to $H$ in $L^{2} (d t \otimes P)$ . Then $\lim_{n \to \infty} I_{T} (H^{n}) = \lim_{n \to \infty} I_{T} (K^{n}) in L^{2} (P),$ so the integral $\int_{0}^{T} H d B$ of Ito integral for square-integrable predictable processes does not depend on the approximating sequence. If $H^{'}$ is any predictable process with $H^{'} = H$ $(d t \otimes P)$ -almost everywhere and finite energy, then $\int_{0}^{T} H^{'} d B = \int_{0}^{T} H d B$ almost surely; the integral is a function of the $(d t \otimes P)$ -class alone.

Facts & Assumptions

Given: AC, the standing hypothesis (H), a finite-energy predictable $H$ , two elementary approximating sequences $H^{n} \to H$ and $K^{n} \to H$ in $L^{2} (d t \otimes P)$ , and a predictable $H^{'}$ with $H^{'} = H$ almost everywhere and finite energy.

[F1]

For elementary predictable $G, G^{'}$ on a common refinement, $I_{T} (G) - I_{T} (G^{'}) = I_{T} (G - G^{'})$ identically and $E (I_{T} (G) - I_{T} (G^{'}))^{2} = E \int_{0}^{T} (G - G^{'})^{2} d s = ∥ G - G^{'} ∥_{L^{2} (d t \otimes P)}^{2}$ . Ito isometry for elementary integrands Ito integral of an elementary predictable process

[F2]

Each of the sequences $I_{T} (H^{n})$ and $I_{T} (K^{n})$ is Cauchy in the complete space $L^{2} (P)$ and therefore has an $L^{2} (P)$ limit. The construction designates the limit obtained from one admissible approximating sequence as $\int_{0}^{T} H d B$ ; equality with the limit from every other sequence is what is proved below. Ito integral for square-integrable predictable processes

[F3]

The $L^{2}$ triangle inequality and the identities $∥ U - V ∥_{2} = 0 ⟺ U = V$ almost surely hold on the quotient space. The $L^{p}$ norm descends to the quotient and makes $L^{p}$ a normed space for $1 \leq p \leq \infty$

[F4]

The elementary integral of the difference is the difference of the elementary integrals on a common refinement, and the class of a bounded elementary integrand depends only on its $(d t \otimes P)$ -class. Ito integral of an elementary predictable process Elementary Ito integrals do not depend on step representation

[F5]

AC is declared for the ambient interfaces; the argument below only uses the two given sequences and the metric algebra of $L^{2}$ . The Axiom of Choice

Proof

technique · direct

1.1

Pass to a common refinement of the two elementary representations at each index and use [F1]: $∥ I_{T} (H^{n}) - I_{T} (K^{n}) ∥_{2}^{2} = ∥ H^{n} - K^{n} ∥_{L^{2} (d t \otimes P)}^{2}$ , a finite quantity depending only on the two indices.

F1given

1.2

The triangle inequality gives $∥ H^{n} - K^{n} ∥_{2} \leq ∥ H^{n} - H ∥_{2} + ∥ H - K^{n} ∥_{2} \to 0$ as $n \to \infty$ , because both approximating sequences converge to $H$ .

F3given

1.3

If $H^{'} = H$ almost everywhere, then every approximating sequence for $H$ is an approximating sequence for $H^{'}$ : $∥ H^{n} - H^{'} ∥_{2} \leq ∥ H^{n} - H ∥_{2} + ∥ H - H^{'} ∥_{2} = ∥ H^{n} - H ∥_{2} \to 0$ , and likewise in the other direction.

F3given

2.1

By step 1.1 and step 1.2 the difference of the two sequences of elementary integrals converges to $0$ in $L^{2} (P)$ , while by [F2] each sequence converges; two convergent sequences with difference tending to $0$ have the same limit, and by [F3] the limits agree as $L^{2} (P)$ -classes, that is, almost surely.

F2F3step 1.1step 1.2

3.1

Consequently the class $\int_{0}^{T} H d B$ is independent of the approximating sequence; and by step 1.3, replacing $H$ by an almost-everywhere equal $H^{'}$ keeps the same admissible sequences and hence the same limit, so $\int_{0}^{T} H^{'} d B = \int_{0}^{T} H d B$ almost surely. Only the two given sequences are used, and AC enters only as the declared ambient interface [F5].

F4F5step 1.3step 2.1∎

Source notes

Van der Vaart, Definition 5.25 and Theorem 5.26, performs the same two descents: independence of the approximating sequence for a fixed integrand, and invariance under changing the integrand on a product-null set. Both are isometry statements for elementary differences, so no pathwise argument is involved.

Depends on

Used by

Deterministic Ito integrals are Gaussian Corollary
Product-measure equality is not pointwise equality Counterexample
Ito isometry and linearity in predictable L2 Theorem
The Ito integral process has a continuous martingale version Theorem

Dependency tree · two levels

37 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Aad van der Vaart, Stochastic Integration and Differential Equations, Definition 5.25 and Theorem 5.26 (standard reference, not scraped)