Weighted Brownian Increment Second Moment

Proved

LevyStochCalc.Brownian.Ito.integral_factor_increment_sq_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

This states a weighted second-moment identity for a Brownian motion $W$ under a probability measure $P$ and a Brownian filtration $\mathcal F$ . Let $0\le a<b$ , and let $g:\Omega\to\mathbb R$ be strongly measurable with respect to $\mathcal F_a$ . Then

\int_\Omega g(\omega)\bigl(W_b(\omega)-W_a(\omega)\bigr)^2\,dP(\omega) =\left(\int_\Omega g\,dP\right)(b-a).

The result records the conditional diagonal calculation for an increment square; its stated assumptions include only strong $\mathcal F_a$ -measurability of $g$ , besides the probability and Brownian-filtration hypotheses.

Original Lean declaration header

Within namespace LevyStochCalc.Brownian.Ito. Shown in its original module context. Proof omitted.

lemma integral_factor_increment_sq
    {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
    (W : LevyStochCalc.Brownian.BrownianMotion P)
    (ℱ : Filtration ℝ ‹MeasurableSpace Ω›) (hℱ : IsBrownianFiltration W ℱ) {a b : ℝ} (ha : 0 ≤ a)
      (hab : a < b)
    {g : Ω → ℝ}
    (hg_meas : @MeasureTheory.StronglyMeasurable Ω ℝ _
      (ℱ a) g) :
    ∫ ω, g ω * (W.W b ω - W.W a ω) ^ 2 ∂P = (∫ ω, g ω ∂P) * (b - a)

Module navigation

Module: LevyStochCalc/Brownian/ItoL2CompletionConvergence.

Definition modules: LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Construction.

Previous: condExp_sq_increment_of_martingale.

Source excerpt, lines 540–548 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoL2CompletionConvergence

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/ItoL2CompletionConvergence.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory
universe u
variable {Ω : Type u} [MeasurableSpace Ω]

Formal statement


lemma LevyStochCalc.Brownian.Ito.integral_factor_increment_sq_readable
    {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
    (W : LevyStochCalc.Brownian.BrownianMotion P)
    (ℱ : Filtration ℝ ‹MeasurableSpace Ω›) (hℱ : IsBrownianFiltration W ℱ) {a b : ℝ} (ha : 0 ≤ a)
      (hab : a < b)
    {g : Ω → ℝ}
    (hg_meas : @MeasureTheory.StronglyMeasurable Ω ℝ _
      (ℱ a) g) :
    ∫ ω, g ω * (W.W b ω - W.W a ω) ^ 2 ∂P = (∫ ω, g ω ∂P) * (b - a) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoL2CompletionConvergence.lean#L537-L566 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoL2CompletionConvergence`, declaration `LevyStochCalc.Brownian.Ito.integral_factor_increment_sq`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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