Summed Squared Norm Bound Along a Uniform Grid

Proved

LevyStochCalc.Brownian.Ito.sum_integral_sq_norm_vectorItoIncrement_le_readable

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

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

This is a uniform-grid estimate for squared vector Itô-process increments.

Let $(\Omega,\mathcal F,P)$ , the $d$ -dimensional Brownian motion, coordinate Brownian filtrations, and integrands $H_{mk}$ satisfy the stated measurability, progressive measurability, finite squared extended-integral, and boundedness assumptions with $C\ge0$ . Let the measurable drifts satisfy $B\ge0$ and $|b_m|\le B$ . For $T>0$ , $m\ne0$ , and $t_i=\operatorname{unifGrid}(T,m,i)$ ,

\sum_{i<m}\int_\Omega\|X_{t_{i+1}}-X_{t_i}\|^2dP \le n\left(2B^2T\frac{T}{m}+2d^2C^2T\right).

This estimate shows that the sum of the second moments over the uniform partition remains bounded and is used in Taylor-remainder arguments.

Original Lean declaration header

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

theorem sum_integral_sq_norm_vectorItoIncrement_le (X₀ : Ω → Fin n → ℝ)
    (bdrift : Fin n → Ω → ℝ → ℝ) (hbm : ∀ p, Measurable (Function.uncurry (bdrift p)))
    {B : ℝ} (hB0 : 0 ≤ B) (hB : ∀ (p : Fin n) (ω : Ω) (s : ℝ), |bdrift p ω s| ≤ B)
    {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    ∑ i ∈ Finset.range m, ∫ ω,
        ‖vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift (unifGrid T m (i + 1)) ω
          - vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift (unifGrid T m i) ω‖ ^ 2 ∂P
      ≤ (n : ℝ) * (2 * (B ^ 2 * (T * (T / (m : ℝ)))) + 2 * ((d : ℝ) ^ 2 * (C ^ 2 * T)))

Module navigation

Module: LevyStochCalc/Brownian/VectorItoTaylor.

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

Supporting theorems: integral_sq_norm_vectorItoProcess_sub_le.

Previous: measurable_taylorRemainderNormed · Next: integral_abs_vectorTaylorRemainder_le_affine.

Source excerpt, lines 83–90 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoTaylor

/-
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/VectorItoTaylor.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] {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 < ⊤)
  {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.sum_integral_sq_norm_vectorItoIncrement_le_readable (X₀ : Ω → Fin n → ℝ)
    (bdrift : Fin n → Ω → ℝ → ℝ) (hbm : ∀ p, Measurable (Function.uncurry (bdrift p)))
    {B : ℝ} (hB0 : 0 ≤ B) (hB : ∀ (p : Fin n) (ω : Ω) (s : ℝ), |bdrift p ω s| ≤ B)
    {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    ∑ i ∈ Finset.range m, ∫ ω,
        ‖vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift (unifGrid T m (i + 1)) ω
          - vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift (unifGrid T m i) ω‖ ^ 2 ∂P
      ≤ (n : ℝ) * (2 * (B ^ 2 * (T * (T / (m : ℝ)))) + 2 * ((d : ℝ) ^ 2 * (C ^ 2 * T))) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoTaylor.lean#L80-L108 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoTaylor`, declaration `LevyStochCalc.Brownian.Ito.sum_integral_sq_norm_vectorItoIncrement_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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