Second-Moment Bound for Compensated Square Increments

Proved

LevyStochCalc.Brownian.Ito.integral_sq_quadVarIncrement_le_readable

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

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

Let $H$ satisfy the measurability, progressive-measurability, joint square-integrability, and uniform bound assumptions above, with $C\ge0$ . For $0\le a<b$ , the compensated square increment has an integrable square and

\int \mathrm{quadVarIncrement}_{a,b}(\omega)^2\,dP(\omega) \le \bigl(2(6+\gamma_4)+2\bigr)C^4(b-a)^2,

where $\gamma_4$ is the Lean constant gaussianFourthMoment. The declaration gives the inequality for the second moment; its integrability follows from the same hypotheses.

Original Lean declaration header

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

theorem integral_sq_quadVarIncrement_le {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    ∫ ω, (quadVarIncrement W ℱ hℱ H hm hp hq a b ω) ^ 2 ∂P
      ≤ (2 * (6 + gaussianFourthMoment) + 2) * C ^ 4 * (b - a) ^ 2

Module navigation

Module: LevyStochCalc/Brownian/ItoQuadVarSum.

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

Supporting theorems: compensator_sub_bounds, lintegral_stochasticIntegralBrownian_sub_pow_four_le, memLp_two_sq_increment.

Previous: memLp_two_sq_increment · Next: exists_continuous_modification_stochasticIntegralBrownian.

Source excerpt, lines 268–270 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoQuadVarSum

/-
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/ItoQuadVarSum.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_sq_quadVarIncrement_le_readable {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    ∫ ω, (quadVarIncrement W ℱ hℱ H hm hp hq a b ω) ^ 2 ∂P
      ≤ (2 * (6 + gaussianFourthMoment) + 2) * C ^ 4 * (b - a) ^ 2 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoQuadVarSum.lean#L266-L372 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoQuadVarSum`, declaration `LevyStochCalc.Brownian.Ito.integral_sq_quadVarIncrement_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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