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Event-weighted Ito isometry after a conditioning time

Statement

Assume AC and the Brownian filtration conventions in Continuous Brownian Ito processes. Let $t \geq 0$ , $h > 0$ , let $A \in F_{t}$ , and let $H$ be a real predictable process with finite energy on $[t, t + h]$ and $H_{s} = 0$ for $s \notin (t, t + h]$ . For any coordinate $W^{j}$ of the driving finite-dimensional Brownian motion, $E ⁣ [1_{A} {∣ \int_{t}^{t + h} H_{s} d W_{s}^{j} ∣}^{2}] = E ⁣ [1_{A} \int_{t}^{t + h} ∣ H_{s} ∣^{2} d s] .$

Facts & Assumptions

Given: AC, the Brownian filtered probability space, deterministic $t, h$ , $A \in F_{t}$ , and the predictable square-integrable integrand $H$ supported on $(t, t + h]$ .

[F1]

Continuous Brownian Ito processes: each Brownian coordinate has increments independent of the current filtration, and finite-energy Ito integrals are defined on predictable integrands.

[F2]

Elementary predictable Brownian integrands: an elementary predictable integrand is constant on finitely many intervals $(s_{k}, s_{k + 1}]$ with bounded coefficient measurable at $s_{k}$ .

[F3]

Ito integral of an elementary predictable process: on a common partition the elementary integral is the finite sum of coefficient times Brownian increment, and it is linear in the integrand.

[F4]

Ito integral for square-integrable predictable processes: a finite-energy predictable integrand is approximated in $L^{2} (d s d P)$ by elementary predictable integrands, and its Ito integral is their $L^{2} (P)$ limit.

[F5]

Ito isometry and linearity in predictable L2: for every finite-energy predictable integrand $G$ , $E ∣ \int G d W^{j} ∣^{2} = E \int ∣ G ∣^{2} d s$ , and the extension is linear in $L^{2}$ .

[F6]

Continuous-time filtrations and all-pairs martingales: a continuous-time filtration satisfies $F_{s} \subseteq F_{r}$ whenever $0 \leq s \leq r$ .

[A1]

The Axiom of Choice: AC is part of the Brownian and Ito-integral interfaces [F1], [F4], and [F5].

Proof

Given: The assumptions above.

1.1F2F4given

Choose elementary predictable $H^{n} \to H$ in $L^{2} (d s d P)$ using [F4], and replace each $H^{n}$ by ${\tilde{H}}^{n} = 1_{(t, t + h]} H^{n}$ . Refining its finite partition at $t$ and $t + h$ makes ${\tilde{H}}^{n}$ elementary; because $H$ vanishes off this interval, ${\tilde{H}}^{n} \to H$ in $L^{2} (d s d P)$ .

2.1F1F2F3F5F6step 1.1

Refine the partition of ${\tilde{H}}^{n}$ so that every nonzero cell $(s_{k}, s_{k + 1}]$ has $s_{k} \geq t$ . Its coefficient $ξ_{k}$ is $F_{s_{k}}$ -measurable, and [F6] with $A \in F_{t}$ makes $1_{A} ξ_{k}$ also $F_{s_{k}}$ -measurable. Thus $1_{A} {\tilde{H}}^{n}$ is elementary predictable for the Brownian coordinate in [F1], and the finite-sum definition [F3] gives $\int 1_{A} {\tilde{H}}^{n} d W^{j} = 1_{A} \int {\tilde{H}}^{n} d W^{j}$ . Applying [F5] to $1_{A} {\tilde{H}}^{n}$ yields $E ⁣ [1_{A} {∣ \int {\tilde{H}}^{n} d W^{j} ∣}^{2}] = E ⁣ [1_{A} \int ∣ {\tilde{H}}^{n} ∣^{2} d s] .$

3.1F4F5step 2.1

Multiplication by $1_{A}$ is a contraction in both $L^{2} (P)$ and $L^{2} (d s d P)$ . Hence [F4] and [F5] give $\int {\tilde{H}}^{n} d W^{j} \to \int H d W^{j}$ and $\int 1_{A} {\tilde{H}}^{n} d W^{j} \to \int 1_{A} H d W^{j}$ in $L^{2} (P)$ , while $1_{A} {\tilde{H}}^{n} \to 1_{A} H$ in $L^{2} (d s d P)$ . Passing the finite-sum identity of step 2.1 to these limits gives $\int 1_{A} H d W^{j} = 1_{A} \int H d W^{j}$ in $L^{2} (P)$ .

4.1F5step 3.1algebra

By [F5] and step 3.1, $E ⁣ [1_{A} {∣ \int H d W^{j} ∣}^{2}] = E {∣ \int 1_{A} H d W^{j} ∣}^{2} = E \int ∣ 1_{A} H ∣^{2} d s = E ⁣ [1_{A} \int ∣ H ∣^{2} d s] .$ All terms are finite because $H$ has finite energy, so the asserted identity follows.

5.1A1step 4.1∎

The argument applies to each coordinate separately and uses only finite partitions, $L^{2}$ limits, and the supplied Ito isometry. Its only inherited choice assumption is [A1].

Depends on

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Sources

Gregory F. Lawler, Stochastic Calculus: An Introduction with Applications, Sections 3.2.2-3.2.3 (standard reference, not scraped)
Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, Sections 4.5 and 10.6, printed pp. 127-140 and 356-365 (standard reference, not scraped)