Difference Isometry for Adapted Simple Integrals

Proved

LevyStochCalc.Brownian.Ito.simpleIntegral_diff_isometry_of_adapted_readable

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

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

Let $W$ be a Brownian motion with Brownian filtration $ℱ$ . Let $G₁,G₂$ be simple predictable integrands, possibly with different horizons, and assume all their coefficients are strongly measurable at the relevant filtration times. For $t≥0$ ,

\int_Ω\|I_{G₁}(t,ω)-I_{G₂}(t,ω)\|^2\,dP =\int_Ω\int_{[0,t]}\|G₁(s,ω)-G₂(s,ω)\|^2\,ds\,dP.

Both sides are ENNReal integrals in $[0,∞]$ . The result extends the intermediate-time Itô isometry to simple integrands whose horizons need not coincide.

Original Lean declaration header

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

theorem simpleIntegral_diff_isometry_of_adapted
    {P : Measure Ω} [IsProbabilityMeasure P] (W : LevyStochCalc.Brownian.BrownianMotion P)
    (ℱ : Filtration ℝ ‹MeasurableSpace Ω›) (hℱ : IsBrownianFiltration W ℱ)
    {T₁ T₂ : ℝ} (G₁ : SimplePredictable Ω T₁) (G₂ : SimplePredictable Ω T₂)
    (h₁ : ∀ i : Fin G₁.N, @MeasureTheory.StronglyMeasurable Ω ℝ _
      (ℱ (G₁.partition i.castSucc)) (G₁.ξ i))
    (h₂ : ∀ i : Fin G₂.N, @MeasureTheory.StronglyMeasurable Ω ℝ _
      (ℱ (G₂.partition i.castSucc)) (G₂.ξ i))
    {t : ℝ} (ht : 0 ≤ t) :
    ∫⁻ ω,
        (‖simpleIntegral W G₁ t ω - simpleIntegral W G₂ t ω‖₊ : ℝ≥0∞) ^ 2 ∂P
      = ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) t,
          (‖G₁.eval s ω - G₂.eval s ω‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P

Module navigation

Module: LevyStochCalc/Brownian/ItoAlgebraSimpleIntegrand.

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

Previous: SimplePredictable.lintegral_eval_sq_lt_top · Next: stochasticIntegralBrownian_eval_simple.

Source excerpt, lines 125–137 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoAlgebraSimpleIntegrand

/-
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/ItoAlgebraSimpleIntegrand.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory
open scoped NNReal ENNReal
universe u
variable {Ω : Type u} [MeasurableSpace Ω]

Formal statement


theorem LevyStochCalc.Brownian.Ito.simpleIntegral_diff_isometry_of_adapted_readable
    {P : Measure Ω} [IsProbabilityMeasure P] (W : LevyStochCalc.Brownian.BrownianMotion P)
    (ℱ : Filtration ℝ ‹MeasurableSpace Ω›) (hℱ : IsBrownianFiltration W ℱ)
    {T₁ T₂ : ℝ} (G₁ : SimplePredictable Ω T₁) (G₂ : SimplePredictable Ω T₂)
    (h₁ : ∀ i : Fin G₁.N, @MeasureTheory.StronglyMeasurable Ω ℝ _
      (ℱ (G₁.partition i.castSucc)) (G₁.ξ i))
    (h₂ : ∀ i : Fin G₂.N, @MeasureTheory.StronglyMeasurable Ω ℝ _
      (ℱ (G₂.partition i.castSucc)) (G₂.ξ i))
    {t : ℝ} (ht : 0 ≤ t) :
    ∫⁻ ω,
        (‖simpleIntegral W G₁ t ω - simpleIntegral W G₂ t ω‖₊ : ℝ≥0∞) ^ 2 ∂P
      = ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) t,
          (‖G₁.eval s ω - G₂.eval s ω‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoAlgebraSimpleIntegrand.lean#L123-L166 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoAlgebraSimpleIntegrand`, declaration `LevyStochCalc.Brownian.Ito.simpleIntegral_diff_isometry_of_adapted`; 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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