Third Absolute Moment of a Scalar Itô-Integral Increment

Proved

LevyStochCalc.Brownian.Ito.integral_abs_sub_pow_three_le_readable

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

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

This formalizes a third-moment estimate for a scalar Itô-integral increment.

Let $P$ be a probability measure, let $W$ be a Brownian motion with Brownian filtration $\mathcal F$ , and let $H:\Omega\times\mathbb R\to\mathbb R$ be measurable and progressively measurable. Assume its squared extended integral on every $[0,T]$ , $T>0$ , is finite. Let $C\ge0$ and suppose $|H(\omega,s)|\le C$ . For $0\le a<b$ , define $c=\texttt{gaussianFourthMoment}$ . Then the increment is cubically integrable and

\int_\Omega\left|\int_a^b H_s\,dW_s\right|^3dP \le \frac{C^2+(6+c)C^4}{2}\,(b-a)\sqrt{b-a}.

This is the scalar third absolute-moment input for vector martingale and Itô-process estimates.

Formalization Note The declaration also asserts integrability of the cubed increment. gaussianFourthMoment is the Lean constant used for the Gaussian fourth moment.

Original Lean declaration header

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

theorem integral_abs_sub_pow_three_le {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    MeasureTheory.Integrable (fun ω => |stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
        - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω| ^ 3) P
      ∧ ∫ ω, |stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
          - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω| ^ 3 ∂P
        ≤ (C ^ 2 + (6 + gaussianFourthMoment) * C ^ 4) / 2
          * ((b - a) * Real.sqrt (b - a))

Module navigation

Module: LevyStochCalc/Brownian/ItoIncrementMomentBounds.

Definition modules: LevyStochCalc/Brownian/ItoL2CompletionMaster, LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Construction, LevyStochCalc/Brownian/ItoFourthMomentIncrements, LevyStochCalc/Probability/Progressive.

Supporting theorems: integral_sub_pow_four_le, integral_sub_sq_le.

Previous: integral_sub_sq_le.

Source excerpt, lines 486–492 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoIncrementMomentBounds

/-
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/ItoIncrementMomentBounds.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)
  (H : Ω → ℝ → ℝ) (hm : Measurable (Function.uncurry H))
  (hp : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ H)
  (hq : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
  {C : ℝ} (hC0 : 0 ≤ C) (hCH : ∀ ω s, |H ω s| ≤ C)
include hℱ hC0 hCH

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_abs_sub_pow_three_le_readable {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    MeasureTheory.Integrable (fun ω => |stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
        - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω| ^ 3) P
      ∧ ∫ ω, |stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
          - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω| ^ 3 ∂P
        ≤ (C ^ 2 + (6 + gaussianFourthMoment) * C ^ 4) / 2
          * ((b - a) * Real.sqrt (b - a)) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoIncrementMomentBounds.lean#L483-L547 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoIncrementMomentBounds`, declaration `LevyStochCalc.Brownian.Ito.integral_abs_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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