Prove2Me

Weighted Polarized Increment Sum Bound

Proved
LevyStochCalc.Brownian.Ito.integral_sq_weighted_polarQuadVarSum_le_readable

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

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

Let t:N→Rt:\mathbb N\to\mathbb R satisfy 0≤t00\le t_0 and tk<tk+1t_k<t_{k+1}. Let gk:Ω→Rg_k:\Omega\to\mathbb R be strongly F.rightConttk\mathcal F.\mathrm{rightCont}_{t_k}-measurable, with D≥0D\ge0 and ∣gk∣≤D|g_k|\le D. Under the stated hypotheses on H1,H2H_1,H_2 and C≥0C\ge0, for every N∈NN\in\mathbb N,

∫(∑i<Ngi polarQuadVarIncrementti,ti+1(H1,H2))2dP≤D2 3(2(6+γ4)+2)(2C)4∑i<N(ti+1−ti)2.\int\left(\sum_{i<N}g_i\,\mathrm{polarQuadVarIncrement}_{t_i,t_{i+1}}(H_1,H_2)\right)^2dP \le D^2\,3\bigl(2(6+\gamma_4)+2\bigr)(2C)^4 \sum_{i<N}(t_{i+1}-t_i)^2.

This is the finite-grid second-moment estimate for conditionally centered polarized increments.

Original Lean declaration header

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

theorem integral_sq_weighted_polarQuadVarSum_le
    (t : ℕ → ℝ) (h0 : 0 ≤ t 0) (ht : ∀ k, t k < t (k + 1))
    (g : ℕ → Ω → ℝ)
    (hg : ∀ k, @MeasureTheory.StronglyMeasurable Ω ℝ _ (ℱ.rightCont (t k)) (g k))
    {D : ℝ} (hD0 : 0 ≤ D) (hgD : ∀ (k : ℕ) (ω : Ω), |g k ω| ≤ D) (N : ℕ) :
    ∫ ω, (∑ i ∈ Finset.range N, g i ω
        * polarQuadVarIncrement W ℱ hℱ H₁ H₂ hm₁ hm₂ hp₁ hp₂ hq₁ hq₂ (t i) (t (i + 1)) ω) ^ 2 ∂P
      ≤ D ^ 2 * (3 * ((2 * (6 + gaussianFourthMoment) + 2) * (2 * C) ^ 4))
        * ∑ i ∈ Finset.range N, (t (i + 1) - t i) ^ 2

Module navigation

Module: LevyStochCalc/Brownian/PolarisedQuadVar.

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

Supporting theorems: condExp_polarQuadVarIncrement, integral_sq_polarQuadVarIncrement_le, memLp_two_polarQuadVarIncrement, stronglyMeasurable_polarQuadVarIncrement, integral_sq_sum_of_condExp_eq_zero.

Previous: integral_sq_polarQuadVarIncrement_le.

Source excerpt, lines 317–325 · 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 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_weighted_polarQuadVarSum_le_readable
    (t : ℕ → ℝ) (h0 : 0 ≤ t 0) (ht : ∀ k, t k < t (k + 1))
    (g : ℕ → Ω → ℝ)
    (hg : ∀ k, @MeasureTheory.StronglyMeasurable Ω ℝ _ (ℱ.rightCont (t k)) (g k))
    {D : ℝ} (hD0 : 0 ≤ D) (hgD : ∀ (k : ℕ) (ω : Ω), |g k ω| ≤ D) (N : ℕ) :
    ∫ ω, (∑ i ∈ Finset.range N, g i ω
        * polarQuadVarIncrement W ℱ hℱ H₁ H₂ hm₁ hm₂ hp₁ hp₂ hq₁ hq₂ (t i) (t (i + 1)) ω) ^ 2 ∂P
      ≤ D ^ 2 * (3 * ((2 * (6 + gaussianFourthMoment) + 2) * (2 * C) ^ 4))
        * ∑ i ∈ Finset.range N, (t (i + 1) - t i) ^ 2 := by sorry
Source
Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/PolarisedQuadVar.lean#L313-L404 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.PolarisedQuadVar`, declaration `LevyStochCalc.Brownian.Ito.integral_sq_weighted_polarQuadVarSum_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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