Negation of the Itô Integral

Proved

LevyStochCalc.Brownian.Ito.stochasticIntegralBrownian_neg_readable

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

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

This proves linearity of the Brownian Itô integral under negation.

Let $(\Omega,\mathcal A,P)$ be a probability space, let $W$ be a Brownian motion, and let $\mathcal F$ be a Brownian filtration. For an integrand $H:\Omega\times\mathbb R\to\mathbb R$ , assume both $H$ and $-H$ are jointly measurable, progressively measurable, and have finite ENNReal $L^2$ energy on every positive interval:

\int_\Omega\int_{[0,T]}\|H(\omega,s)\|_+^2\,ds\,dP<\top, \qquad \int_\Omega\int_{[0,T]}\|-H(\omega,s)\|_+^2\,ds\,dP<\top.

For every $T>0$ , the integral of the negated integrand equals the negation of the integral, $P$ -almost everywhere:

I_T[-H](\omega)=-I_T[H](\omega).

This is the negation case used to derive subtractive and finite-sum linearity.

Original Lean declaration header

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

theorem stochasticIntegralBrownian_neg {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 < ⊤)
    (hmn : Measurable (Function.uncurry fun ω s => -H ω s))
    (hpn : Probability.ProgressivelyMeasurable ℱ fun ω s => -H ω s)
    (hqn : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖-H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {T : ℝ} (hT : 0 < T) :
    stochasticIntegralBrownian W ℱ hℱ (fun ω s => -H ω s) hmn hpn hqn T
      =ᵐ[P] fun ω => -stochasticIntegralBrownian W ℱ hℱ H hm hp hq T ω

Module navigation

Module: LevyStochCalc/Brownian/ItoFinsetSum.

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

Supporting theorems: stochasticIntegralBrownian_const_mul.

Next: stochasticIntegralBrownian_sub.

Source excerpt, lines 48–58 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoFinsetSum

/-
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/ItoFinsetSum.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] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
variable (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.stochasticIntegralBrownian_neg_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 < ⊤)
    (hmn : Measurable (Function.uncurry fun ω s => -H ω s))
    (hpn : Probability.ProgressivelyMeasurable ℱ fun ω s => -H ω s)
    (hqn : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖-H ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {T : ℝ} (hT : 0 < T) :
    stochasticIntegralBrownian W ℱ hℱ (fun ω s => -H ω s) hmn hpn hqn T
      =ᵐ[P] fun ω => -stochasticIntegralBrownian W ℱ hℱ H hm hp hq T ω := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoFinsetSum.lean#L46-L70 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoFinsetSum`, declaration `LevyStochCalc.Brownian.Ito.stochasticIntegralBrownian_neg`; 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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