Conditional Centering of Polarized Increments

Proved

LevyStochCalc.Brownian.Ito.condExp_polarQuadVarIncrement_readable

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

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

Let $H_1,H_2$ be two measurable, progressively measurable integrands satisfying the stated finite joint-square-integral conditions, and assume their sum is also measurable, progressively measurable, and jointly square-integrable. If $C\ge0$ and both $|H_1|$ and $|H_2|$ are bounded by $C$ , then for $0\le a<b$ ,

\mathbb E\!\left[\mathrm{polarQuadVarIncrement}_{a,b}(H_1,H_2)\mid\mathcal F.\mathrm{rightCont}_a\right]=0\quad\text{a.e. }P.

The polarized increment is the product of the two Itô-integral increments compensated by the time integral of $H_1H_2$ .

Original Lean declaration header

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

theorem condExp_polarQuadVarIncrement {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    P[polarQuadVarIncrement W ℱ hℱ H₁ H₂ hm₁ hm₂ hp₁ hp₂ hq₁ hq₂ a b | ℱ.rightCont a]
      =ᵐ[P] 0

Module navigation

Module: LevyStochCalc/Brownian/PolarisedQuadVar.

Definition modules: LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Construction, LevyStochCalc/Probability/Progressive.

Supporting theorems: compensator_add_sq_eq, compensator_sub_bounds, condExp_quadVarIncrement, memLp_two_sq_increment, stochasticIntegralBrownian_add.

Previous: memLp_two_polarQuadVarIncrement · Next: stronglyMeasurable_polarQuadVarIncrement.

Source excerpt, lines 171–173 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_PolarisedQuadVar

/-
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/PolarisedQuadVar.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]
variable {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] (W : LevyStochCalc.Brownian.BrownianMotion P)
  (ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›) (hℱ : LevyStochCalc.Brownian.IsBrownianFiltration W ℱ)
  (H₁ H₂ : Ω → ℝ → ℝ)
  (hm₁ : Measurable (Function.uncurry H₁)) (hm₂ : Measurable (Function.uncurry H₂))
  (hp₁ : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ H₁)
  (hp₂ : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ H₂)
  (hq₁ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H₁ ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
  (hq₂ : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
  (hma : Measurable (Function.uncurry fun ω s => H₁ ω s + H₂ ω s))
  (hpa : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ fun ω s => H₁ ω s + H₂ ω s)
  (hqa : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H₁ ω s + H₂ ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
variable {C : ℝ} (hC0 : 0 ≤ C) (hCH₁ : ∀ ω s, |H₁ ω s| ≤ C) (hCH₂ : ∀ ω s, |H₂ ω s| ≤ C)
include hℱ hma hpa hqa hC0 hCH₁ hCH₂

Formal statement


theorem LevyStochCalc.Brownian.Ito.condExp_polarQuadVarIncrement_readable {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    P[polarQuadVarIncrement W ℱ hℱ H₁ H₂ hm₁ hm₂ hp₁ hp₂ hq₁ hq₂ a b | ℱ.rightCont a]
      =ᵐ[P] 0 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/PolarisedQuadVar.lean#L169-L204 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.PolarisedQuadVar`, declaration `LevyStochCalc.Brownian.Ito.condExp_polarQuadVarIncrement`; 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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