Drift–Drift Cell-Sum Bound

Proved

LevyStochCalc.Brownian.Ito.IsVectorItoVersion.integral_abs_sum_drift_sq_le_readable

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

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

This is an $L^1$ estimate for a weighted product of two drift cell integrals.

Let $X$ be a vector Itô-process version with measurable drift coordinates bounded by $B\ge0$ . Let $p,q\in\mathrm{Fin}(n)$ , let $\phi$ be continuous and satisfy $|\phi(x)|\le K_\phi$ with $K_\phi\ge0$ , and let $T>0$ and $m\ne0$ . Then the cell sum is integrable and

\int\left|\sum_{i<m}\phi(X_{t_i}) \left(\int_{(t_i,t_{i+1}]}b_p\,ds\right) \left(\int_{(t_i,t_{i+1}]}b_q\,ds\right)\right|dP \le mK_\phi\left(B\frac{T}{m}\right)^2,

where $t_i=\texttt{unifGrid}(T,m,i)$ .

This is the $O(1/m)$ drift–drift term used in quadratic-variation approximations.

Original Lean declaration header

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

theorem IsVectorItoVersion.integral_abs_sum_drift_sq_le
    (h : IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X)
    (hbm : ∀ m, Measurable (Function.uncurry (bdrift m))) {B : ℝ} (hB0 : 0 ≤ B)
    (hB : ∀ (m : Fin n) (ω : Ω) (s : ℝ), |bdrift m ω s| ≤ B) (p q : Fin n)
    {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)
        * ((∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)), bdrift p ω s ∂volume)
          * ∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)),
              bdrift q ω s ∂volume)) P
      ∧ ∫ ω, |∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
          * ((∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)), bdrift p ω s ∂volume)
            * ∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)),
                bdrift q ω s ∂volume)| ∂P
        ≤ (m : ℝ) * (Kφ * (B * (T / (m : ℝ))) ^ 2)

Module navigation

Module: LevyStochCalc/Brownian/VectorItoQuadVarRiemannTermBounds.

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

Supporting theorems: integral_abs_weighted_cellSum_le, abs_setIntegral_Ioc_le.

Previous: integral_abs_weighted_cellSum_le · Next: IsVectorItoVersion.integral_abs_sum_driftCross_le.

Source excerpt, lines 135–149 · 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)

Formal statement


theorem LevyStochCalc.Brownian.Ito.IsVectorItoVersion.integral_abs_sum_drift_sq_le_readable
    (h : IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X)
    (hbm : ∀ m, Measurable (Function.uncurry (bdrift m))) {B : ℝ} (hB0 : 0 ≤ B)
    (hB : ∀ (m : Fin n) (ω : Ω) (s : ℝ), |bdrift m ω s| ≤ B) (p q : Fin n)
    {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)
        * ((∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)), bdrift p ω s ∂volume)
          * ∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)),
              bdrift q ω s ∂volume)) P
      ∧ ∫ ω, |∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
          * ((∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)), bdrift p ω s ∂volume)
            * ∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)),
                bdrift q ω s ∂volume)| ∂P
        ≤ (m : ℝ) * (Kφ * (B * (T / (m : ℝ))) ^ 2) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoQuadVarRiemannTermBounds.lean#L134-L184 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoQuadVarRiemannTermBounds`, declaration `LevyStochCalc.Brownian.Ito.IsVectorItoVersion.integral_abs_sum_drift_sq_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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