Continuous Adapted Modification of the Itô Integral

Proved

LevyStochCalc.Brownian.Ito.exists_continuousAdapted_modification_readable

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

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

This theorem constructs a continuous adapted version of the bounded-integrand Itô integral. In addition to the Brownian, filtration, measurability, progressive-measurability, energy, and bound hypotheses, assume that $\mathcal F_0\le\mathcal F_t$ for $t\le0$ and that every $P$ -null measurable set is measurable in $\mathcal F_0$ . Then there exists $Y:\mathbb R\to\Omega\to\mathbb R$ such that

\forall\omega,\;t\mapsto Y(t,\omega)\text{ is continuous},\qquad \forall u,\;Y(u)\text{ is }\mathcal F_u\text{-measurable},

and for every $t\ge0$ , $Y(t)$ equals the stochastic integral of $H$ up to $t$ almost surely. The null-set assumptions are part of the statement.

Original Lean declaration header

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

theorem exists_continuousAdapted_modification
    (hℱ0 : ∀ t : ℝ, t ≤ 0 → ℱ 0 ≤ ℱ t)
    (hnull : ∀ s : Set Ω, MeasurableSet s → P s = 0 → MeasurableSet[ℱ 0] s) :
    ∃ Y : ℝ → Ω → ℝ, (∀ ω : Ω, Continuous fun t => Y t ω)
      ∧ (∀ u : ℝ, Measurable[ℱ u] (Y u))
      ∧ ∀ t : ℝ, 0 ≤ t → Y t =ᵐ[P] stochasticIntegralBrownian W ℱ hℱ H hm hp hq t

Module navigation

Module: LevyStochCalc/Brownian/ItoVersionExists.

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

Supporting theorems: exists_continuous_modification_stochasticIntegralBrownian.

Source excerpt, lines 119–124 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoVersionExists

/-
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/ItoVersionExists.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)
  (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.exists_continuousAdapted_modification_readable
    (hℱ0 : ∀ t : ℝ, t ≤ 0 → ℱ 0 ≤ ℱ t)
    (hnull : ∀ s : Set Ω, MeasurableSet s → P s = 0 → MeasurableSet[ℱ 0] s) :
    ∃ Y : ℝ → Ω → ℝ, (∀ ω : Ω, Continuous fun t => Y t ω)
      ∧ (∀ u : ℝ, Measurable[ℱ u] (Y u))
      ∧ ∀ t : ℝ, 0 ≤ t → Y t =ᵐ[P] stochasticIntegralBrownian W ℱ hℱ H hm hp hq t := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoVersionExists.lean#L116-L200 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoVersionExists`, declaration `LevyStochCalc.Brownian.Ito.exists_continuousAdapted_modification`; 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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