L¹ Bound for Diagonal Polarized Grid Sums

Proved

LevyStochCalc.Brownian.Ito.IsVectorItoVersion.integral_abs_sum_polarQuadVar_le_readable

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

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

Consider a $d$ -dimensional Brownian motion, a vector Itô process $X$ with $n$ coordinates, and coordinate integrands bounded in absolute value by $C\ge0$ . Let $p,q\in\mathrm{Fin}(n)$ and $k\in\mathrm{Fin}(d)$ , and assume the required measurability, progressive-measurability, integrability, and sum-integrand hypotheses. If $\phi$ is continuous, $|\phi(x)|\le K_\phi$ with $K_\phi\ge0$ , $T>0$ , and $m\ne0$ , then the weighted diagonal sum is integrable and

\mathbb E\left|\sum_{i<m}\phi(X_{t_i})\,\mathrm{polarQuadVarIncrement}_{t_i,t_{i+1}}(H^{p,k},H^{q,k})\right| \le\sqrt{K_\phi^2\,3\bigl(2(6+\gamma_4)+2\bigr)(2C)^4\frac{T^2}{m}},

where $t_i=\mathrm{unifGrid}(T,m,i)$ . The conclusion is an $L^1$ bound of order $m^{-1/2}$ .

Original Lean declaration header

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

theorem IsVectorItoVersion.integral_abs_sum_polarQuadVar_le
    (h : IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X) (p q : Fin n) (k : Fin d)
    (hma : Measurable (Function.uncurry fun ω s => H p k ω s + H q k ω s))
    (hpa : Probability.ProgressivelyMeasurable ℱ fun ω s => H p k ω s + H q k ω s)
    (hqa : ∀ T', 0 < T' → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T',
      (‖H p k ω s + H q k ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {φ : (Fin n → ℝ) → ℝ} (hφc : Continuous φ) {Kφ : ℝ} (hKφ0 : 0 ≤ Kφ)
    (hφbd : ∀ x, |φ x| ≤ Kφ) {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    Integrable (fun ω : Ω => ∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
        * polarQuadVarIncrement (W.W k) ℱ (hcoord k) (H p k) (H q k) (hHm p k) (hHm q k)
            (hHp p k) (hHp q k) (hHs p k) (hHs q k)
            (unifGrid T m i) (unifGrid T m (i + 1)) ω) P
      ∧ ∫ ω, |∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
          * polarQuadVarIncrement (W.W k) ℱ (hcoord k) (H p k) (H q k) (hHm p k) (hHm q k)
              (hHp p k) (hHp q k) (hHs p k) (hHs q k)
              (unifGrid T m i) (unifGrid T m (i + 1)) ω| ∂P
        ≤ Real.sqrt (Kφ ^ 2 * (3 * ((2 * (6 + gaussianFourthMoment) + 2) * (2 * C) ^ 4))
          * (T ^ 2 / (m : ℝ)))

Module navigation

Module: LevyStochCalc/Brownian/VectorItoQuadVarRiemannTermBounds.

Definition modules: LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Multidim, LevyStochCalc/Brownian/ItoFormulaGrid, LevyStochCalc/Brownian/ItoFourthMomentIncrements, LevyStochCalc/Probability/Progressive, LevyStochCalc/Brownian/PolarisedQuadVar, LevyStochCalc/Brownian/VectorItoProcessVersion.

Supporting theorems: integral_sq_weighted_polarQuadVarSum_le, memLp_two_polarQuadVarIncrement, stronglyMeasurable_polarQuadVarIncrement, integral_abs_le_sqrt_of_integral_sq_le.

Previous: IsVectorItoVersion.integral_abs_sum_driftCross_le · Next: IsVectorItoVersion.integral_abs_sum_offDiagCross_le.

Source excerpt, lines 282–299 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoQuadVarRiemannTermBounds

/-
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/VectorItoQuadVarRiemannTermBounds.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] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {n d : ℕ} {W : LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion P d}
  {ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›}
  {hcoord : ∀ k : Fin d, LevyStochCalc.Brownian.IsBrownianFiltration (W.W k) ℱ}
  {H : Fin n → Fin d → Ω → ℝ → ℝ}
  {hHm : ∀ m k, Measurable (Function.uncurry (H m k))}
  {hHp : ∀ m k, LevyStochCalc.Probability.ProgressivelyMeasurable ℱ (H m k)}
  {hHs : ∀ (m : Fin n) (k : Fin d) (T : ℝ), 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H m k ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤}
  {X₀ : Ω → Fin n → ℝ} {bdrift : Fin n → Ω → ℝ → ℝ} {X : ℝ → Ω → Fin n → ℝ}
  {C : ℝ} (hC0 : 0 ≤ C)
  (hCH : ∀ (m : Fin n) (k : Fin d) (ω : Ω) (s : ℝ), |H m k ω s| ≤ C)
include hC0 hCH

Formal statement


theorem LevyStochCalc.Brownian.Ito.IsVectorItoVersion.integral_abs_sum_polarQuadVar_le_readable
    (h : IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X) (p q : Fin n) (k : Fin d)
    (hma : Measurable (Function.uncurry fun ω s => H p k ω s + H q k ω s))
    (hpa : Probability.ProgressivelyMeasurable ℱ fun ω s => H p k ω s + H q k ω s)
    (hqa : ∀ T', 0 < T' → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T',
      (‖H p k ω s + H q k ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {φ : (Fin n → ℝ) → ℝ} (hφc : Continuous φ) {Kφ : ℝ} (hKφ0 : 0 ≤ Kφ)
    (hφbd : ∀ x, |φ x| ≤ Kφ) {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    Integrable (fun ω : Ω => ∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
        * polarQuadVarIncrement (W.W k) ℱ (hcoord k) (H p k) (H q k) (hHm p k) (hHm q k)
            (hHp p k) (hHp q k) (hHs p k) (hHs q k)
            (unifGrid T m i) (unifGrid T m (i + 1)) ω) P
      ∧ ∫ ω, |∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
          * polarQuadVarIncrement (W.W k) ℱ (hcoord k) (H p k) (H q k) (hHm p k) (hHm q k)
              (hHp p k) (hHp q k) (hHs p k) (hHs q k)
              (unifGrid T m i) (unifGrid T m (i + 1)) ω| ∂P
        ≤ Real.sqrt (Kφ ^ 2 * (3 * ((2 * (6 + gaussianFourthMoment) + 2) * (2 * C) ^ 4))
          * (T ^ 2 / (m : ℝ))) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoQuadVarRiemannTermBounds.lean#L280-L335 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoQuadVarRiemannTermBounds`, declaration `LevyStochCalc.Brownian.Ito.IsVectorItoVersion.integral_abs_sum_polarQuadVar_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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