Subtractive Linearity of the Itô Integral

Proved

LevyStochCalc.Brownian.Ito.stochasticIntegralBrownian_sub_readable

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

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

This proves subtractive linearity of the Brownian Itô integral.

Let $(\Omega,\mathcal A,P)$ be a probability space with Brownian motion $W$ and Brownian filtration $\mathcal F$ . Let $H_1,H_2:\Omega\times\mathbb R\to\mathbb R$ . Assume each of $H_1$ , $H_2$ , and $H_1-H_2$ is jointly measurable, progressively measurable, and has finite ENNReal energy on every positive horizon:

\int_\Omega\int_{[0,T]}\|H_i(\omega,s)\|_+^2\,ds\,dP<\top \quad(i=1,2),

with the analogous condition for $H_1-H_2$ .

For every $T>0$ ,

I_T[H_1-H_2](\omega) =I_T[H_1](\omega)-I_T[H_2](\omega)

for $P$ -almost every $\omega$ .

The result is a basic linearity rule for later finite-sum constructions.

Original Lean declaration header

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

theorem stochasticIntegralBrownian_sub {H₁ H₂ : Ω → ℝ → ℝ}
    (hm₁ : Measurable (Function.uncurry H₁)) (hm₂ : Measurable (Function.uncurry H₂))
    (hp₁ : Probability.ProgressivelyMeasurable ℱ H₁)
    (hp₂ : Probability.ProgressivelyMeasurable ℱ H₂)
    (hq₁ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₁ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    (hq₂ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    (hms : Measurable (Function.uncurry fun ω s => H₁ ω s - H₂ ω s))
    (hps : Probability.ProgressivelyMeasurable ℱ fun ω s => H₁ ω s - H₂ ω s)
    (hqs : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₁ ω s - H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {T : ℝ} (hT : 0 < T) :
    stochasticIntegralBrownian W ℱ hℱ (fun ω s => H₁ ω s - H₂ ω s) hms hps hqs T
      =ᵐ[P] fun ω => stochasticIntegralBrownian W ℱ hℱ H₁ hm₁ hp₁ hq₁ T ω
        - stochasticIntegralBrownian W ℱ hℱ H₂ hm₂ hp₂ hq₂ 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_add, stochasticIntegralBrownian_neg.

Previous: stochasticIntegralBrownian_neg · Next: exists_stochasticIntegralBrownian_finsetSum.

Source excerpt, lines 74–89 · 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_sub_readable {H₁ H₂ : Ω → ℝ → ℝ}
    (hm₁ : Measurable (Function.uncurry H₁)) (hm₂ : Measurable (Function.uncurry H₂))
    (hp₁ : Probability.ProgressivelyMeasurable ℱ H₁)
    (hp₂ : Probability.ProgressivelyMeasurable ℱ H₂)
    (hq₁ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₁ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    (hq₂ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    (hms : Measurable (Function.uncurry fun ω s => H₁ ω s - H₂ ω s))
    (hps : Probability.ProgressivelyMeasurable ℱ fun ω s => H₁ ω s - H₂ ω s)
    (hqs : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₁ ω s - H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {T : ℝ} (hT : 0 < T) :
    stochasticIntegralBrownian W ℱ hℱ (fun ω s => H₁ ω s - H₂ ω s) hms hps hqs T
      =ᵐ[P] fun ω => stochasticIntegralBrownian W ℱ hℱ H₁ hm₁ hp₁ hq₁ T ω
        - stochasticIntegralBrownian W ℱ hℱ H₂ hm₂ hp₂ hq₂ T ω := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoFinsetSum.lean#L72-L120 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoFinsetSum`, declaration `LevyStochCalc.Brownian.Ito.stochasticIntegralBrownian_sub`; 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).

View graph