Almost-Everywhere Quadratic-Variation Riemann Decomposition

Proved

LevyStochCalc.Brownian.Ito.IsVectorItoVersion.quadVarRiemann_decomp_ae_readable

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

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

Let $X$ be a vector Itô process with bounded Brownian coefficients $|H^{m,k}|\le C$ , measurable bounded drift $|b^m|\le B$ , and the stated progressive-measurability and integrability assumptions. For coordinate indices $p,q$ , a function $\phi$ , $T>0$ , and $m\ne0$ , the weighted product sum over the uniform grid equals almost everywhere the sum of six groups: drift–drift terms, two drift–martingale terms, diagonal compensated products, off-diagonal Brownian cross terms, and the frozen Riemann sum

\sum_{i<m}\phi(X_{t_i})\int_{(t_i,t_{i+1}]} \sum_k H^{p,k}_sH^{q,k}_s\,ds.

The equality is $P$ -almost everywhere and uses quadVarRiemann_decomp_ae_readable exactly in that scope.

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.

                - vectorItoMartingale W ℱ hcoord H hHm hHp hHs p (unifGrid T m i) ω)))
        + (∑ k : Fin d, ∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
            * polarQuadVarIncrement (W.W k) ℱ (hcoord k) (H p k) (H q k) (hHm p k) (hHm q k)
                (hHp p k) (hHp q k) (hHs p k) (hHs q k)
                (unifGrid T m i) (unifGrid T m (i + 1)) ω)
        + (∑ k : Fin d, ∑ l ∈ Finset.univ.erase k, ∑ i ∈ Finset.range m,
            φ (X (unifGrid T m i) ω)
              * crossIncrement W hcoord (hHm p k) (hHp p k) (hHs p k) (hHm q l) (hHp q l)
                  (hHs q l) k l (unifGrid T m i) (unifGrid T m (i + 1)) ω)
        + ∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
            * ∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)),
                ∑ k : Fin d, H p k ω s * H q k ω s ∂volume

Module navigation

Module: LevyStochCalc/Brownian/VectorItoQuadVarRiemannDecomposition.

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

Supporting theorems: vectorItoMartingale_prod_eq.

Previous: vectorItoMartingale_prod_eq · Next: IsVectorItoVersion.integral_abs_quadVarRiemann_sub_le.

Source excerpt, lines 181–192 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoQuadVarRiemannDecomposition

/-
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/VectorItoQuadVarRiemannDecomposition.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 : ∀ m k, Measurable (Function.uncurry (H m k))}
  {hHp : ∀ m k, LevyStochCalc.Probability.ProgressivelyMeasurable ℱ (H m k)}
  {hHs : ∀ (m : Fin n) (k : Fin d) (T : ℝ), 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H m 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}
  {C : ℝ} (hC0 : 0 ≤ C)
  (hCH : ∀ (m : Fin n) (k : Fin d) (ω : Ω) (s : ℝ), |H m k ω s| ≤ C)
include hC0 hCH

Formal statement


theorem LevyStochCalc.Brownian.Ito.IsVectorItoVersion.quadVarRiemann_decomp_ae_readable
    (h : IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X)
    (hbm : ∀ m, Measurable (Function.uncurry (bdrift m))) {B : ℝ}
    (hB : ∀ (m : Fin n) (ω : Ω) (s : ℝ), |bdrift m ω s| ≤ B) (p q : Fin n)
    (φ : (Fin n → ℝ) → ℝ) {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    ∀ᵐ ω ∂P, (∑ i ∈ Finset.range m, φ (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)))
      = (∑ i ∈ Finset.range m, φ (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 (unifGrid T m i) (unifGrid T m (i + 1)), bdrift q ω s ∂volume))
        + (∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
            * ((∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)), bdrift p ω s ∂volume)
              * (vectorItoMartingale W ℱ hcoord H hHm hHp hHs q (unifGrid T m (i + 1)) ω
                - vectorItoMartingale W ℱ hcoord H hHm hHp hHs q (unifGrid T m i) ω)))
        + (∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
            * ((∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)), bdrift q ω s ∂volume)
              * (vectorItoMartingale W ℱ hcoord H hHm hHp hHs p (unifGrid T m (i + 1)) ω
                - vectorItoMartingale W ℱ hcoord H hHm hHp hHs p (unifGrid T m i) ω)))
        + (∑ k : Fin d, ∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
            * polarQuadVarIncrement (W.W k) ℱ (hcoord k) (H p k) (H q k) (hHm p k) (hHm q k)
                (hHp p k) (hHp q k) (hHs p k) (hHs q k)
                (unifGrid T m i) (unifGrid T m (i + 1)) ω)
        + (∑ k : Fin d, ∑ l ∈ Finset.univ.erase k, ∑ i ∈ Finset.range m,
            φ (X (unifGrid T m i) ω)
              * crossIncrement W hcoord (hHm p k) (hHp p k) (hHs p k) (hHm q l) (hHp q l)
                  (hHs q l) k l (unifGrid T m i) (unifGrid T m (i + 1)) ω)
        + ∑ i ∈ Finset.range m, φ (X (unifGrid T m i) ω)
            * ∫ s in Set.Ioc (unifGrid T m i) (unifGrid T m (i + 1)),
                ∑ 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/VectorItoQuadVarRiemannDecomposition.lean#L161-L256 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoQuadVarRiemannDecomposition`, declaration `LevyStochCalc.Brownian.Ito.IsVectorItoVersion.quadVarRiemann_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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