Third Moment Bound for a Vector Itô-Process Norm

Proved

LevyStochCalc.Brownian.Ito.integral_norm_vectorItoProcess_sub_pow_three_le_readable

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

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

This is a third absolute-moment estimate for the norm of a vector Itô-process increment.

Let $(\Omega,\mathcal F,P)$ be a probability space, let $W$ be a $d$ -dimensional Brownian motion with coordinate Brownian filtrations, and let the integrands $H_{mk}$ satisfy measurability, progressive measurability, finite squared extended-integral, and $|H_{mk}|\le C$ hypotheses with $C\ge0$ . Let vectorItoProcess be defined from $X_0$ and measurable drifts $b_m$ , with $B\ge0$ and $|b_m(\omega,s)|\le B$ . For $0\le u<v$ ,

\int_\Omega\|X_v-X_u\|^3dP \le n^2\,n\,4\left((B(v-u))^3+d^2\,d\,\frac{C^2+(6+\texttt{gaussianFourthMoment})C^4}{2} (v-u)\sqrt{v-u}\right).

This is the vector norm version of the coordinatewise third-moment estimate and is the main cubic increment bound for Taylor remainders.

Original Lean declaration header

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

theorem integral_norm_vectorItoProcess_sub_pow_three_le (X₀ : Ω → Fin n → ℝ)
    (bdrift : Fin n → Ω → ℝ → ℝ) (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)
    {u v : ℝ} (hu : 0 ≤ u) (huv : u < v) :
    ∫ ω, ‖vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift v ω
        - vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift u ω‖ ^ 3 ∂P
      ≤ (n : ℝ) ^ 2 * ((n : ℝ) * (4 * ((B * (v - u)) ^ 3
        + (d : ℝ) ^ 2 * ((d : ℝ) * ((C ^ 2 + (6 + gaussianFourthMoment) * C ^ 4) / 2
          * ((v - u) * Real.sqrt (v - u)))))))

Module navigation

Module: LevyStochCalc/Brownian/VectorItoThirdMoment.

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

Selected supporting theorems: integrable_abs_drift_pow, integrable_abs_sum_pow_three, integral_abs_sub_pow_three_le, integral_abs_vectorItoProcess_sub_pow_three_le, integral_norm_pi_pow_three_le.

Previous: integral_abs_vectorItoProcess_sub_pow_three_le · Next: sum_integral_norm_vectorItoIncrement_pow_three_le.

Source excerpt, lines 169–177 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoThirdMoment

/-
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/VectorItoThirdMoment.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.integral_norm_vectorItoProcess_sub_pow_three_le_readable (X₀ : Ω → Fin n → ℝ)
    (bdrift : Fin n → Ω → ℝ → ℝ) (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)
    {u v : ℝ} (hu : 0 ≤ u) (huv : u < v) :
    ∫ ω, ‖vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift v ω
        - vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift u ω‖ ^ 3 ∂P
      ≤ (n : ℝ) ^ 2 * ((n : ℝ) * (4 * ((B * (v - u)) ^ 3
        + (d : ℝ) ^ 2 * ((d : ℝ) * ((C ^ 2 + (6 + gaussianFourthMoment) * C ^ 4) / 2
          * ((v - u) * Real.sqrt (v - u))))))) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoThirdMoment.lean#L167-L193 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoThirdMoment`, declaration `LevyStochCalc.Brownian.Ito.integral_norm_vectorItoProcess_sub_pow_three_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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