Second-Moment Bound for a Bounded Itô-Integral Increment

Proved

LevyStochCalc.Brownian.Ito.lintegral_stochasticIntegralBrownian_sub_sq_le_readable

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

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

This proves the second-moment estimate for an increment of a scalar Brownian Itô integral.

Let $(\Omega,\mathcal A,P)$ be a probability space, let $W$ be a Brownian motion, and let $\mathcal F$ be a Brownian filtration. Let $H:\Omega\times\mathbb R\to\mathbb R$ be jointly measurable and progressively measurable, with finite ENNReal energy on every positive horizon. Suppose that for some $C\ge0$ ,

|H(\omega,s)|\le C

for every $\omega,s$ . Let $0\le a<b$ . Then

\int_\Omega \left\|I_b[H](\omega)-I_a[H](\omega)\right\|_+^2\,dP \le \operatorname{ofReal}\!\left(C^2(b-a)\right).

The left side is an ENNReal integral, and the right side is the ENNReal.ofReal coercion of the real quantity $C^2(b-a)$ . This is the basic variance-type increment estimate used in subsequent moment bounds.

Original Lean declaration header

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

theorem lintegral_stochasticIntegralBrownian_sub_sq_le
    (H : Ω → ℝ → ℝ) (hm : Measurable (Function.uncurry H))
    (hp : Probability.ProgressivelyMeasurable ℱ H)
    (hq : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {C : ℝ} (hC0 : 0 ≤ C) (hCH : ∀ ω s, |H ω s| ≤ C)
    {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    ∫⁻ ω, (‖stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
        - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω‖₊ : ℝ≥0∞) ^ 2 ∂P
      ≤ ENNReal.ofReal (C ^ 2 * (b - a))

Module navigation

Module: LevyStochCalc/Brownian/ItoIncrementMomentBounds.

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

Supporting theorems: stochasticIntegralBrownian_indicator_Ioc.

Previous: lintegral_stochasticIntegralBrownian_sub_pow_four_le · Next: integral_sub_pow_four_le.

Source excerpt, lines 292–301 · 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 MeasureTheory ProbabilityTheory
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)
include hℱ

Formal statement


theorem LevyStochCalc.Brownian.Ito.lintegral_stochasticIntegralBrownian_sub_sq_le_readable
    (H : Ω → ℝ → ℝ) (hm : Measurable (Function.uncurry H))
    (hp : Probability.ProgressivelyMeasurable ℱ H)
    (hq : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {C : ℝ} (hC0 : 0 ≤ C) (hCH : ∀ ω s, |H ω s| ≤ C)
    {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    ∫⁻ ω, (‖stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
        - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω‖₊ : ℝ≥0∞) ^ 2 ∂P
      ≤ ENNReal.ofReal (C ^ 2 * (b - a)) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoIncrementMomentBounds.lean#L289-L355 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoIncrementMomentBounds`, declaration `LevyStochCalc.Brownian.Ito.lintegral_stochasticIntegralBrownian_sub_sq_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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