Continuous Modification of a Bounded-Integrand Itô Integral

Proved

LevyStochCalc.Brownian.Ito.exists_continuous_modification_stochasticIntegralBrownian_readable

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

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

This theorem gives a continuous modification of the real Itô integral driven by Brownian motion. Let $W$ be a Brownian motion with filtration $\mathcal F$ , let $H:\Omega\times\mathbb R\to\mathbb R$ be measurable and progressively measurable, and assume its squared norm has finite ENNReal integral on every interval $[0,T]$ with $T>0$ . If $|H(\omega,s)|\le C$ for all $\omega,s$ and $C\ge0$ , then there exists $Y:\mathbb R\to\Omega\to\mathbb R$ such that

P\text{-a.e. }\omega,\quad t\mapsto Y(t,\omega)\text{ is continuous},

and for every $t\ge0$ , $Y(t)$ equals the stochastic integral of $H$ up to time $t$ almost surely. The result uses the fourth-moment Kolmogorov condition with exponents $(p,q)=(4,2)$ .

Original Lean declaration header

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

theorem exists_continuous_modification_stochasticIntegralBrownian :
    ∃ Y : ℝ → Ω → ℝ,
      (∀ᵐ ω ∂P, Continuous fun t => Y t ω)
        ∧ ∀ t : ℝ, 0 ≤ t →
            Y t =ᵐ[P] stochasticIntegralBrownian W ℱ hℱ H hm hp hq t

Module navigation

Module: LevyStochCalc/Brownian/ItoQuadVarSum.

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

Supporting theorems: kolmogorovChentsov_modification, lintegral_stochasticIntegralBrownian_sub_pow_four_le.

Previous: integral_sq_quadVarIncrement_le.

Source excerpt, lines 523–527 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoQuadVarSum

/-
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/ItoQuadVarSum.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
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_continuous_modification_stochasticIntegralBrownian_readable :
    ∃ Y : ℝ → Ω → ℝ,
      (∀ᵐ ω ∂P, Continuous fun t => Y t ω)
        ∧ ∀ 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/ItoQuadVarSum.lean#L520-L599 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoQuadVarSum`, declaration `LevyStochCalc.Brownian.Ito.exists_continuous_modification_stochasticIntegralBrownian`; 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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