Integrability and Fourth-Moment Bound for an Itô Increment

Proved

LevyStochCalc.Brownian.Ito.integral_sub_pow_four_le_readable

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

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

This theorem establishes both integrability and a real fourth-moment bound for an Itô-integral increment.

Let $W$ be a Brownian motion with Brownian filtration $\mathcal F$ , let $H$ be jointly measurable and progressively measurable, and assume its squared norm has finite $[0,\infty]$ -valued integral on every positive finite time interval. Suppose $C\ge0$ , $|H(\omega,s)|\le C$ , and $0\le a<b$ . Then the function

\omega\mapsto\left(\int_0^b H_s\,dW_s-\int_0^a H_s\,dW_s\right)^4

is integrable with respect to $P$ , and

\int\left(\int_0^b H_s\,dW_s-\int_0^a H_s\,dW_s\right)^4dP \le (6+\mathsf g_4)C^4(b-a)^2.

This converts the preceding ENNReal fourth-moment estimate into an ordinary integral statement and explicitly supplies integrability.

Original Lean declaration header

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

theorem integral_sub_pow_four_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) ^ 4) P
      ∧ ∫ ω, (stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
          - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω) ^ 4 ∂P
        ≤ (6 + gaussianFourthMoment) * C ^ 4 * (b - a) ^ 2

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: lintegral_stochasticIntegralBrownian_sub_pow_four_le.

Previous: lintegral_stochasticIntegralBrownian_sub_sq_le · Next: integral_sub_sq_le.

Source excerpt, lines 398–403 · 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_sub_pow_four_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) ^ 4) P
      ∧ ∫ ω, (stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
          - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω) ^ 4 ∂P
        ≤ (6 + gaussianFourthMoment) * C ^ 4 * (b - a) ^ 2 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoIncrementMomentBounds.lean#L396-L439 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoIncrementMomentBounds`, declaration `LevyStochCalc.Brownian.Ito.integral_sub_pow_four_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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