Local Congruence of Itô Integrals

Proved

LevyStochCalc.Brownian.Ito.stochasticIntegralBrownian_congr_of_le_readable

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

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

This proves that two integrands agreeing before a stopping time yield the same integral before that stopping time.

Let $\tau$ be a stopping time for a filtration $\mathcal F$ supporting a Brownian motion $W$ . Let $H_1,H_2:\Omega\times\mathbb R\to\mathbb R$ be jointly measurable and progressively measurable, and assume that each has finite ENNReal energy on every positive horizon. Suppose that for every $\omega$ and every $s>0$ ,

s\le\tau(\omega)\quad\Longrightarrow\quad H_1(\omega,s)=H_2(\omega,s).

Then for every $t>0$ , for $P$ -almost every $\omega$ satisfying $t\le\tau(\omega)$ ,

I_t[H_1](\omega)=I_t[H_2](\omega).

This is the principal reuse theorem for replacing an integrand by an equivalent one on the portion of time before a stopping time.

Original Lean declaration header

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

theorem stochasticIntegralBrownian_congr_of_le
    (hτ : MeasureTheory.IsStoppingTime ℱ τ)
    {H₁ H₂ : Ω → ℝ → ℝ} (hm₁ : Measurable (Function.uncurry H₁))
    (hp₁ : Probability.ProgressivelyMeasurable ℱ H₁)
    (hq₁ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₁ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    (hm₂ : Measurable (Function.uncurry H₂))
    (hp₂ : Probability.ProgressivelyMeasurable ℱ H₂)
    (hq₂ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    (hagree : ∀ (ω : Ω) (s : ℝ), 0 < s → ((s : ℝ) : WithTop ℝ) ≤ τ ω → H₁ ω s = H₂ ω s)
    {t : ℝ} (ht : 0 < t) :
    ∀ᵐ ω ∂P, ((t : ℝ) : WithTop ℝ) ≤ τ 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/ItoLocality.

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

Supporting theorems: stochasticIntegralBrownian_congr_of_pos, stochasticIntegralBrownian_stopped_eq_of_le.

Previous: stochasticIntegralBrownian_congr_of_pos.

Source excerpt, lines 566–580 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoLocality

/-
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/ItoLocality.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal Topology
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  (τ : Ω → WithTop ℝ)
variable (W : LevyStochCalc.Brownian.BrownianMotion P)
  (ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›) (hℱ : LevyStochCalc.Brownian.IsBrownianFiltration W ℱ)

Formal statement


theorem LevyStochCalc.Brownian.Ito.stochasticIntegralBrownian_congr_of_le_readable
    (hτ : MeasureTheory.IsStoppingTime ℱ τ)
    {H₁ H₂ : Ω → ℝ → ℝ} (hm₁ : Measurable (Function.uncurry H₁))
    (hp₁ : Probability.ProgressivelyMeasurable ℱ H₁)
    (hq₁ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₁ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    (hm₂ : Measurable (Function.uncurry H₂))
    (hp₂ : Probability.ProgressivelyMeasurable ℱ H₂)
    (hq₂ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    (hagree : ∀ (ω : Ω) (s : ℝ), 0 < s → ((s : ℝ) : WithTop ℝ) ≤ τ ω → H₁ ω s = H₂ ω s)
    {t : ℝ} (ht : 0 < t) :
    ∀ᵐ ω ∂P, ((t : ℝ) : WithTop ℝ) ≤ τ 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/ItoLocality.lean#L564-L600 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoLocality`, declaration `LevyStochCalc.Brownian.Ito.stochasticIntegralBrownian_congr_of_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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