Almost-Everywhere Grid Decomposition of Vector Itô Formula

Proved

LevyStochCalc.Brownian.Ito.IsVectorItoVersion.vectorItoFormula_decomp_ae_readable

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

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

This is the grid decomposition of the vector Itô formula.

Let $(\Omega,\mathcal F,P)$ be a probability space, let $W$ be a $d$ -dimensional Brownian motion with coordinate Brownian filtrations, and let $H_{pk}$ satisfy the stated measurability, progressive measurability, and finite squared extended-integral assumptions. Let $X$ be an IsVectorItoVersion with measurable drifts $b_p$ bounded by $B$ , and let $f,f',f''$ be a function and first and second derivative data. Assume that each process $\partial_p f'(X_s)H_{pk}(s)$ is measurable, progressively measurable, and has finite squared extended integral on every positive interval. Let $T>0$ and $m\ne0$ .

The defect of the continuous-time expression equals, $P$ -almost everywhere, the Taylor remainder plus the drift, martingale, and quadratic-variation grid errors:

\begin{aligned} &f(X_T)-f(X_0)-\Bigg[\sum_p\int_{(0,T]}\partial_p f'(X_s)b_p(s)\,ds +\sum_{p,k}\int_0^T\partial_p f'(X_s)H_{pk}(s)\,dW_s^k\\ &\qquad\qquad+\frac12\sum_{p,q}\int_{(0,T]}\partial_{pq}f''(X_s)\sum_kH_{pk}(s)H_{qk}(s)\,ds\Bigg] \\ ={}&\text{Taylor remainder on the grid}\\ &+\text{drift grid sum minus drift integral}\\ &+\text{martingale grid sum minus stochastic integral}\\ &+\text{quadratic-variation grid sum minus its time integral}. \end{aligned}

The four terms on the right are exactly those defined in the Lean statement through taylorRemainderNormed, coordDeriv, coordDeriv₂, coordItoIntegral, and stochasticIntegralBrownian.

This declaration packages the algebraic decomposition that precedes estimates and convergence arguments for the vector Itô formula.

Formalization Note The equality is an a.e. equality under $P$ ; the grid is unifGrid, and all time integrals use the interval $(0,T]$ .

Original Lean conclusion excerpt

Within namespace LevyStochCalc.Brownian.Ito. Shown in its original module context. Earlier hypotheses and the proof are omitted from this excerpt; the complete verified statement remains below.

                * (coordItoIntegral W ℱ hcoord H hHm hHp hHs p k (unifGrid T m (i + 1)) ω
                  - coordItoIntegral W ℱ hcoord H hHm hHp hHs p k (unifGrid T m i) ω))
              - stochasticIntegralBrownian (W.W k) ℱ (hcoord k)
                  (fun ω s => coordDeriv f' p (X s ω) * H p k ω s)
                  (hmg p k) (hpg p k) (hqg p k) T ω))
        + 1 / 2 * ∑ p : Fin n, ∑ q : Fin n,
            ((∑ i ∈ Finset.range m, coordDeriv₂ f'' p q (X (unifGrid T m i) ω)
                * ((X (unifGrid T m (i + 1)) ω p - X (unifGrid T m i) ω p)
                  * (X (unifGrid T m (i + 1)) ω q - X (unifGrid T m i) ω q)))
              - ∫ s in Set.Ioc (0 : ℝ) T,
                  coordDeriv₂ f'' p q (X s ω)
                    * ∑ k : Fin d, H p k ω s * H q k ω s ∂volume)

Module navigation

Module: LevyStochCalc/Brownian/VectorItoFormulaDecomp.

Definition modules: LevyStochCalc/Brownian/ItoL2CompletionMaster, LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Multidim, LevyStochCalc/Brownian/CoordDerivative, LevyStochCalc/Brownian/ItoFormulaGrid, LevyStochCalc/Probability/Progressive, LevyStochCalc/Brownian/TaylorTwoVector, LevyStochCalc/Brownian/VectorItoProcess, LevyStochCalc/Brownian/VectorItoProcessVersion.

Supporting theorems: sum_apply_fderiv_eq, sum_apply_fderiv₂_eq.

Previous: sum_apply_fderiv_eq.

Source excerpt, lines 196–207 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoFormulaDecomp

/-
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/VectorItoFormulaDecomp.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 < ⊤}
  {X₀ : Ω → Fin n → ℝ} {bdrift : Fin n → Ω → ℝ → ℝ} {X : ℝ → Ω → Fin n → ℝ}

Formal statement


theorem LevyStochCalc.Brownian.Ito.IsVectorItoVersion.vectorItoFormula_decomp_ae_readable
    (h : IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X)
    (hbm : ∀ p, Measurable (Function.uncurry (bdrift p))) {B : ℝ}
    (hB : ∀ (p : Fin n) (ω : Ω) (s : ℝ), |bdrift p ω s| ≤ B)
    {f : (Fin n → ℝ) → ℝ} {f' : (Fin n → ℝ) → (Fin n → ℝ) →L[ℝ] ℝ}
    {f'' : (Fin n → ℝ) → (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) →L[ℝ] ℝ}
    (hmg : ∀ (p : Fin n) (k : Fin d),
      Measurable (Function.uncurry fun ω s => coordDeriv f' p (X s ω) * H p k ω s))
    (hpg : ∀ (p : Fin n) (k : Fin d), Probability.ProgressivelyMeasurable ℱ
      fun ω s => coordDeriv f' p (X s ω) * H p k ω s)
    (hqg : ∀ (p : Fin n) (k : Fin d) (T' : ℝ), 0 < T' → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T',
      (‖coordDeriv f' p (X s ω) * H p k ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    (fun ω : Ω => f (X T ω) - f (X 0 ω)
        - ((∑ p : Fin n, ∫ s in Set.Ioc (0 : ℝ) T,
              coordDeriv f' p (X s ω) * bdrift p ω s ∂volume)
          + (∑ p : Fin n, ∑ k : Fin d, stochasticIntegralBrownian (W.W k) ℱ (hcoord k)
              (fun ω s => coordDeriv f' p (X s ω) * H p k ω s)
              (hmg p k) (hpg p k) (hqg p k) T ω)
          + 1 / 2 * ∑ p : Fin n, ∑ q : Fin n, ∫ s in Set.Ioc (0 : ℝ) T,
              coordDeriv₂ f'' p q (X s ω) * ∑ k : Fin d, H p k ω s * H q k ω s ∂volume))
      =ᵐ[P] fun ω : Ω =>
        taylorRemainderNormed f f' f'' (fun i => X (unifGrid T m i) ω) m
        + (∑ p : Fin n, ((∑ i ∈ Finset.range m, coordDeriv f' p (X (unifGrid T m i) ω)
              * ∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)), bdrift p ω s ∂volume)
            - ∫ s in Set.Ioc (0 : ℝ) T, coordDeriv f' p (X s ω) * bdrift p ω s ∂volume))
        + (∑ p : Fin n, ∑ k : Fin d,
            ((∑ i ∈ Finset.range m, coordDeriv f' p (X (unifGrid T m i) ω)
                * (coordItoIntegral W ℱ hcoord H hHm hHp hHs p k (unifGrid T m (i + 1)) ω
                  - coordItoIntegral W ℱ hcoord H hHm hHp hHs p k (unifGrid T m i) ω))
              - stochasticIntegralBrownian (W.W k) ℱ (hcoord k)
                  (fun ω s => coordDeriv f' p (X s ω) * H p k ω s)
                  (hmg p k) (hpg p k) (hqg p k) T ω))
        + 1 / 2 * ∑ p : Fin n, ∑ q : Fin n,
            ((∑ i ∈ Finset.range m, coordDeriv₂ f'' p q (X (unifGrid T m i) ω)
                * ((X (unifGrid T m (i + 1)) ω p - X (unifGrid T m i) ω p)
                  * (X (unifGrid T m (i + 1)) ω q - X (unifGrid T m i) ω q)))
              - ∫ s in Set.Ioc (0 : ℝ) T,
                  coordDeriv₂ f'' p q (X s ω)
                    * ∑ k : Fin d, H p k ω s * H q k ω s ∂volume) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoFormulaDecomp.lean#L165-L220 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoFormulaDecomp`, declaration `LevyStochCalc.Brownian.Ito.IsVectorItoVersion.vectorItoFormula_decomp_ae`; 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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