Conditional Centering of Cross Increments

Proved

LevyStochCalc.Brownian.Ito.condExp_mul_cross_increment_eq_zero_readable

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

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

This is a technical conditional-centering statement for cross increments of Itô integrals driven by distinct Brownian coordinates.

Let $(\Omega,\mathcal A,P)$ be a probability space, let $\mathcal F$ be a filtration, and let $W$ be a $d$ -dimensional Brownian motion whose coordinates are Brownian motions relative to $\mathcal F$ . Let $H,K:\Omega\times\mathbb R\to\mathbb R$ be jointly measurable and progressively measurable, and assume that for every $T>0$ their squared norms have finite $[0,\infty]$ -valued integral over $\Omega\times[0,T]$ . For distinct coordinates $i,j\in\operatorname{Fin}(d)$ and times $0\le a<b$ , the conditional expectation at time $a$ of the product of the two Itô-integral increments is almost everywhere zero:

\mathbb E\!\left[\left(\int_a^b H_s\,dW^i_s\right) \left(\int_a^b K_s\,dW^j_s\right)\middle|\mathcal F_a\right]=0 \quad P\text{-a.e.}

Here the conditional expectation is the Lean definition condExp, and the stochastic integrals are stochasticIntegralBrownian; the hypotheses include the stated measurability, progressive measurability, and finite squared-norm integrability conditions for both integrands. The result is reusable whenever cross-coordinate Itô increments must be treated as martingale differences.

Original Lean declaration header

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

theorem condExp_mul_cross_increment_eq_zero {i j : Fin d} (hij : i ≠ j) {a b : ℝ} (ha : 0 ≤ a)
    (hab : a < b) :
    P[fun ω => (stochasticIntegralBrownian (W.W i) ℱ (hcoord i) H hHm hHp hHs b ω
          - stochasticIntegralBrownian (W.W i) ℱ (hcoord i) H hHm hHp hHs a ω)
        * (stochasticIntegralBrownian (W.W j) ℱ (hcoord j) K hKm hKp hKs b ω
          - stochasticIntegralBrownian (W.W j) ℱ (hcoord j) K hKm hKp hKs a ω) | ℱ a]
      =ᵐ[P] 0

Module navigation

Module: LevyStochCalc/Brownian/CrossVariation.

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

Supporting theorems: integral_mul_cross_increment_eq_zero.

Previous: integral_mul_cross_increment_eq_zero.

Source excerpt, lines 269–275 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_CrossVariation

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Modified 2026-10-12 by the KLS formalization project: ported to Lean v4.33.0 and
-- Mathlib db584cd. The changes are listed in vendor/README.md and vendor/port.patch.
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/CrossVariation.lean; Apache-2.0; locally patched, see formalization/vendor/port.patch; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›}
variable {d : ℕ} (W : LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion P d)
  (hcoord : ∀ k : Fin d, LevyStochCalc.Brownian.IsBrownianFiltration (W.W k) ℱ)
  {H K : Ω → ℝ → ℝ} (hHm : Measurable (Function.uncurry H))
  (hHp : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ H)
  (hHs : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
  (hKm : Measurable (Function.uncurry K))
  (hKp : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ K)
  (hKs : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖K ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)

Formal statement


theorem LevyStochCalc.Brownian.Ito.condExp_mul_cross_increment_eq_zero_readable {i j : Fin d} (hij : i ≠ j) {a b : ℝ} (ha : 0 ≤ a)
    (hab : a < b) :
    P[fun ω => (stochasticIntegralBrownian (W.W i) ℱ (hcoord i) H hHm hHp hHs b ω
          - stochasticIntegralBrownian (W.W i) ℱ (hcoord i) H hHm hHp hHs a ω)
        * (stochasticIntegralBrownian (W.W j) ℱ (hcoord j) K hKm hKp hKs b ω
          - stochasticIntegralBrownian (W.W j) ℱ (hcoord j) K hKm hKp hKs a ω) | ℱ a]
      =ᵐ[P] 0 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/CrossVariation.lean (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.CrossVariation`, declaration `LevyStochCalc.Brownian.Ito.condExp_mul_cross_increment_eq_zero`; the repository is private at upload time); vendored copy patched by formalization/vendor/port.patch (project file formalization/vendor/LevyStochCalc/LevyStochCalc/Brownian/CrossVariation.lean, lines 266-307). 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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