Coordinate Decomposition of the First-Order Taylor Sum

Proved

LevyStochCalc.Brownian.Ito.sum_apply_fderiv_eq_readable

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

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

This is the first-order Taylor-sum decomposition for a vector Itô-process version.

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 measurability, progressive measurability, and finite squared extended-integral assumptions. Let $X$ be an IsVectorItoVersion with measurable drifts $b_p$ . Assume that the drifts are bounded in absolute value by $B\in\mathbb R$ (without a separate nonnegativity assumption), let $f'$ be a continuous linear map-valued function on $\operatorname{Fin}(n)\to\mathbb R$ , and let $T>0$ and $m\ne0$ .

For almost every $\omega$ ,

\begin{aligned} \sum_{i<m} f'(X_{t_i})\bigl(X_{t_{i+1}}-X_{t_i}\bigr) ={}&\sum_p\sum_{i<m}\partial_p f'(X_{t_i})\int_{(t_i,t_{i+1}]} b_p(\omega,s)\,ds\\ &+\sum_p\sum_k\sum_{i<m}\partial_p f'(X_{t_i}) \bigl(I_{pk}(t_{i+1},\omega)-I_{pk}(t_i,\omega)\bigr), \end{aligned}

where $t_i=\operatorname{unifGrid}(T,m,i)$ and $I_{pk}$ is coordItoIntegral.

This identity isolates the drift and martingale Riemann sums in the vector Itô decomposition.

Formalization Note The equality is almost everywhere with respect to $P$ ; coordDeriv, coordItoIntegral, and unifGrid are the relevant Lean definitions.

Original Lean declaration header

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

theorem sum_apply_fderiv_eq
    (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) → (Fin n → ℝ) →L[ℝ] ℝ) {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    ∀ᵐ ω ∂P, ∑ i ∈ Finset.range m, f' (X (unifGrid T m i) ω)
          (X (unifGrid T m (i + 1)) ω - X (unifGrid T m i) ω)
      = (∑ 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)
        + ∑ 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)

Module navigation

Module: LevyStochCalc/Brownian/VectorItoFormulaDecomp.

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

Supporting theorems: sum_range_add_sum_comm.

Previous: sum_apply_fderiv₂_eq · Next: IsVectorItoVersion.vectorItoFormula_decomp_ae.

Source excerpt, lines 122–134 · 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.sum_apply_fderiv_eq_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) → (Fin n → ℝ) →L[ℝ] ℝ) {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    ∀ᵐ ω ∂P, ∑ i ∈ Finset.range m, f' (X (unifGrid T m i) ω)
          (X (unifGrid T m (i + 1)) ω - X (unifGrid T m i) ω)
      = (∑ 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)
        + ∑ 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) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoFormulaDecomp.lean#L118-L163 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoFormulaDecomp`, declaration `LevyStochCalc.Brownian.Ito.sum_apply_fderiv_eq`; 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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