Conditional Centering of Compensated Square Increments

Proved

LevyStochCalc.Brownian.Ito.condExp_quadVarIncrement_readable

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

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

Let $W$ be a Brownian motion with Brownian filtration $\mathcal F$ , and let $H$ be measurable, progressively measurable, and square-integrable on every finite positive horizon in the stated joint sense. For $0\le a\le b$ , define quadVarIncrement as the squared Itô-integral increment minus the compensator increment. Then

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

Thus the compensated square increment is conditionally centered at its left endpoint.

Original Lean declaration header

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

theorem condExp_quadVarIncrement {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) :
    P[quadVarIncrement W ℱ hℱ H hm hp hq a b | ℱ.rightCont a] =ᵐ[P] 0

Module navigation

Module: LevyStochCalc/Brownian/ItoQuadVarSum.

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

Supporting theorems: martingale_rightCont_quadVar_stochasticIntegralBrownian.

Next: compensator_sub_bounds.

Source excerpt, lines 58–59 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoQuadVarSum

/-
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/ItoQuadVarSum.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)
  (H : Ω → ℝ → ℝ) (hm : Measurable (Function.uncurry H))
  (hp : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ H)
  (hq : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
include hℱ

Formal statement


theorem LevyStochCalc.Brownian.Ito.condExp_quadVarIncrement_readable {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) :
    P[quadVarIncrement W ℱ hℱ H hm hp hq a b | ℱ.rightCont a] =ᵐ[P] 0 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoQuadVarSum.lean#L56-L138 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoQuadVarSum`, declaration `LevyStochCalc.Brownian.Ito.condExp_quadVarIncrement`; 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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