Continuous Adapted Vector Itô Version

Proved

LevyStochCalc.Brownian.Ito.exists_isVectorItoVersion_readable

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

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

Let $W$ be a $d$ -dimensional Brownian motion, let $\mathcal F$ satisfy the Brownian-filtration hypotheses, and let $H_{pk}$ be measurable, progressively measurable, square-integrable coefficients with a uniform bound $C$ . Let $X_0:\Omega\to\mathbb R^n$ have measurable coordinates, and let $b_p$ be measurable and progressively measurable with $|b_p(\omega,s)|\le B$ for some $B\ge0$ . Assume the filtration is constant before time $0$ in the stated sense and contains measurable null sets. Then there exists

X:\mathbb R\times\Omega\to(\mathrm{Fin}\,n\to\mathbb R)

which is a continuous adapted version of the vector Itô process IsVectorItoVersion W ℱ hcoord H ... X₀ bdrift X. The theorem asserts existence of the version and does not specify a uniqueness statement.

Original Lean declaration header

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

theorem exists_isVectorItoVersion
    (hℱ0 : ∀ t : ℝ, t ≤ 0 → ℱ 0 ≤ ℱ t)
    (hnull : ∀ s : Set Ω, MeasurableSet s → P s = 0 → MeasurableSet[ℱ 0] s)
    {X₀ : Ω → Fin n → ℝ} (hX₀ : ∀ p : Fin n, Measurable[ℱ 0] fun ω => X₀ ω p)
    (bdrift : Fin n → Ω → ℝ → ℝ) (hbm : ∀ p, Measurable (Function.uncurry (bdrift p)))
    (hbp : ∀ p, Probability.ProgressivelyMeasurable ℱ (bdrift p))
    {B : ℝ} (hB0 : 0 ≤ B) (hB : ∀ (p : Fin n) (ω : Ω) (s : ℝ), |bdrift p ω s| ≤ B) :
    ∃ X : ℝ → Ω → Fin n → ℝ,
      IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X

Module navigation

Module: LevyStochCalc/Brownian/VectorItoVersionExists.

Definition modules: LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Multidim, LevyStochCalc/Probability/Progressive, LevyStochCalc/Brownian/VectorItoProcessVersion.

Supporting theorems: exists_continuousAdapted_modification, abs_setIntegral_Icc_sub_le.

Source excerpt, lines 50–58 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoVersionExists

/-
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/VectorItoVersionExists.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]
  {n d : ℕ} (W : LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion P d)
  (ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›)
  (hcoord : ∀ k : Fin d, LevyStochCalc.Brownian.IsBrownianFiltration (W.W k) ℱ)
  (H : Fin n → Fin d → Ω → ℝ → ℝ)
  (hHm : ∀ p k, Measurable (Function.uncurry (H p k)))
  (hHp : ∀ p k, LevyStochCalc.Probability.ProgressivelyMeasurable ℱ (H p k))
  (hHs : ∀ (p : Fin n) (k : Fin d) (T : ℝ), 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H p k ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
  {C : ℝ} (hC0 : 0 ≤ C)
  (hCH : ∀ (p : Fin n) (k : Fin d) (ω : Ω) (s : ℝ), |H p k ω s| ≤ C)
include hC0 hCH

Formal statement


theorem LevyStochCalc.Brownian.Ito.exists_isVectorItoVersion_readable
    (hℱ0 : ∀ t : ℝ, t ≤ 0 → ℱ 0 ≤ ℱ t)
    (hnull : ∀ s : Set Ω, MeasurableSet s → P s = 0 → MeasurableSet[ℱ 0] s)
    {X₀ : Ω → Fin n → ℝ} (hX₀ : ∀ p : Fin n, Measurable[ℱ 0] fun ω => X₀ ω p)
    (bdrift : Fin n → Ω → ℝ → ℝ) (hbm : ∀ p, Measurable (Function.uncurry (bdrift p)))
    (hbp : ∀ p, Probability.ProgressivelyMeasurable ℱ (bdrift p))
    {B : ℝ} (hB0 : 0 ≤ B) (hB : ∀ (p : Fin n) (ω : Ω) (s : ℝ), |bdrift p ω s| ≤ B) :
    ∃ X : ℝ → Ω → Fin n → ℝ,
      IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoVersionExists.lean#L47-L94 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoVersionExists`, declaration `LevyStochCalc.Brownian.Ito.exists_isVectorItoVersion`; 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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