Factorization of a Brownian Increment Power

Proved

LevyStochCalc.Brownian.Ito.integral_mul_increment_pow_readable

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

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

This is an independence factorization for powers of a Brownian increment.

Let $W$ be a Brownian motion with Brownian filtration $\mathcal F$ . Let $0\le a<b$ , let $Y:\Omega\to\mathbb R$ be strongly measurable with respect to $\mathcal F_a$ and measurable, and let $m\in\mathbb N$ . Then

\int Y(\omega)\bigl(W_b(\omega)-W_a(\omega)\bigr)^m\,dP = \left(\int Y\,dP\right) \left(\int (W_b-W_a)^m\,dP\right).

This statement formalizes the independence of information available by time $a$ from every power of the increment over $(a,b]$ .

Original Lean declaration header

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

theorem integral_mul_increment_pow (ℱ : Filtration ℝ ‹MeasurableSpace Ω›)
    (hℱ : IsBrownianFiltration W ℱ) {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) {Y : Ω → ℝ}
    (hY : @MeasureTheory.StronglyMeasurable Ω ℝ _ (ℱ a) Y) (hYm : Measurable Y) (m : ℕ) :
    ∫ ω, Y ω * (W.W b ω - W.W a ω) ^ m ∂P
      = (∫ ω, Y ω ∂P) * ∫ ω, (W.W b ω - W.W a ω) ^ m ∂P

Module navigation

Module: LevyStochCalc/Brownian/ItoFourthMomentIncrements.

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

Previous: integral_pow_gaussianReal_zero · Next: stronglyMeasurable_partialSum.

Source excerpt, lines 112–116 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoFourthMomentIncrements

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Modified 2026-10-12 by the KLS formalization project: ported to Lean v4.33.0 and
-- Mathlib db584cd. The changes are listed in vendor/README.md and vendor/port.patch.
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/ItoFourthMomentIncrements.lean; Apache-2.0; locally patched, see formalization/vendor/port.patch; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
universe u
variable {Ω : Type u} [MeasurableSpace Ω]
variable {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] (W : LevyStochCalc.Brownian.BrownianMotion P)

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_mul_increment_pow_readable (ℱ : Filtration ℝ ‹MeasurableSpace Ω›)
    (hℱ : IsBrownianFiltration W ℱ) {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) {Y : Ω → ℝ}
    (hY : @MeasureTheory.StronglyMeasurable Ω ℝ _ (ℱ a) Y) (hYm : Measurable Y) (m : ℕ) :
    ∫ ω, Y ω * (W.W b ω - W.W a ω) ^ m ∂P
      = (∫ ω, Y ω ∂P) * ∫ ω, (W.W b ω - W.W a ω) ^ m ∂P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoFourthMomentIncrements.lean (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoFourthMomentIncrements`, declaration `LevyStochCalc.Brownian.Ito.integral_mul_increment_pow`; the repository is private at upload time); vendored copy patched by formalization/vendor/port.patch (project file formalization/vendor/LevyStochCalc/LevyStochCalc/Brownian/ItoFourthMomentIncrements.lean, lines 110-129). 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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