Second-Moment Bound for Polarized Increments

Proved

LevyStochCalc.Brownian.Ito.integral_sq_polarQuadVarIncrement_le_readable

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

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

Under the measurability, progressive-measurability, joint square-integrability, and bounds $C\ge0$ , $|H_1|\le C$ , and $|H_2|\le C$ , assume also the required hypotheses for $H_1+H_2$ . For $0\le a<b$ , the squared polarized compensated increment is integrable and

\int \mathrm{polarQuadVarIncrement}_{a,b}(H_1,H_2)^2\,dP \le 3\bigl(2(6+\gamma_4)+2\bigr)(2C)^4(b-a)^2.

The conclusion includes both integrability and the displayed upper bound.

Original Lean declaration header

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

theorem integral_sq_polarQuadVarIncrement_le {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    Integrable (fun ω =>
        (polarQuadVarIncrement W ℱ hℱ H₁ H₂ hm₁ hm₂ hp₁ hp₂ hq₁ hq₂ a b ω) ^ 2) P
      ∧ ∫ ω, (polarQuadVarIncrement W ℱ hℱ H₁ H₂ hm₁ hm₂ hp₁ hp₂ hq₁ hq₂ a b ω) ^ 2 ∂P
        ≤ 3 * ((2 * (6 + gaussianFourthMoment) + 2) * (2 * C) ^ 4) * (b - a) ^ 2

Module navigation

Module: LevyStochCalc/Brownian/PolarisedQuadVar.

Definition modules: LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Construction, LevyStochCalc/Brownian/ItoFourthMomentIncrements, LevyStochCalc/Probability/Progressive.

Selected supporting theorems: compensator_add_sq_eq, compensator_sub_bounds, integral_sq_quadVarIncrement_le, memLp_two_polarQuadVarIncrement, memLp_two_sq_increment.

Previous: stronglyMeasurable_polarQuadVarIncrement · Next: integral_sq_weighted_polarQuadVarSum_le.

Source excerpt, lines 230–234 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_PolarisedQuadVar

/-
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/PolarisedQuadVar.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal
universe u
variable {Ω : Type u} [MeasurableSpace Ω]
variable {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] (W : LevyStochCalc.Brownian.BrownianMotion P)
  (ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›) (hℱ : LevyStochCalc.Brownian.IsBrownianFiltration W ℱ)
  (H₁ H₂ : Ω → ℝ → ℝ)
  (hm₁ : Measurable (Function.uncurry H₁)) (hm₂ : Measurable (Function.uncurry H₂))
  (hp₁ : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ H₁)
  (hp₂ : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ H₂)
  (hq₁ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H₁ ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
  (hq₂ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
  (hma : Measurable (Function.uncurry fun ω s => H₁ ω s + H₂ ω s))
  (hpa : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ fun ω s => H₁ ω s + H₂ ω s)
  (hqa : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H₁ ω s + H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
variable {C : ℝ} (hC0 : 0 ≤ C) (hCH₁ : ∀ ω s, |H₁ ω s| ≤ C) (hCH₂ : ∀ ω s, |H₂ ω s| ≤ C)
include hℱ hma hpa hqa hC0 hCH₁ hCH₂

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_sq_polarQuadVarIncrement_le_readable {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    Integrable (fun ω =>
        (polarQuadVarIncrement W ℱ hℱ H₁ H₂ hm₁ hm₂ hp₁ hp₂ hq₁ hq₂ a b ω) ^ 2) P
      ∧ ∫ ω, (polarQuadVarIncrement W ℱ hℱ H₁ H₂ hm₁ hm₂ hp₁ hp₂ hq₁ hq₂ a b ω) ^ 2 ∂P
        ≤ 3 * ((2 * (6 + gaussianFourthMoment) + 2) * (2 * C) ^ 4) * (b - a) ^ 2 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/PolarisedQuadVar.lean#L228-L311 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.PolarisedQuadVar`, declaration `LevyStochCalc.Brownian.Ito.integral_sq_polarQuadVarIncrement_le`; 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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