Variance Budget of Tiles Inside an Interval

Proved

LevyStochCalc.Brownian.Ito.SimplePredictable.varClock_le_Ioc_readable

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

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

This theorem bounds the variance clock of tiles contained in an interval.

Let $H$ be a simple predictable integrand with partition times, and let $a\le b$ . Define the per-tile bound to be $C$ when the tile lies inside $(a,b]$ , meaning $a\le\tau_j$ and $\tau_{j+1}\le b$ , and $0$ otherwise. Then

\operatorname{varClock}\!\left(H,\,j\mapsto \begin{cases}C,&a\le\tau_j\text{ and }\tau_{j+1}\le b,\\0,&\text{otherwise,} \end{cases}\!,N\right) \le C^2(b-a).

This estimate is the deterministic localization step behind moment bounds for integrands restricted to $(a,b]$ .

Original Lean declaration header

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

theorem SimplePredictable.varClock_le_Ioc {T : ℝ} (H : SimplePredictable Ω T) {a b C : ℝ}
    (hab : a ≤ b) :
    H.varClock (fun j => if a ≤ H.partition j.castSucc ∧ H.partition j.succ ≤ b then C else 0)
        H.N ≤ C ^ 2 * (b - a)

Module navigation

Module: LevyStochCalc/Brownian/ItoIncrementMomentRestrict.

Definition modules: LevyStochCalc/Brownian/ItoSimpleIntegrand, LevyStochCalc/Brownian/ItoFourthMomentIncrements.

Previous: SimplePredictable.abs_eval_le · Next: SimplePredictable.exists_restrict_Ioc.

Source excerpt, lines 76–79 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoIncrementMomentRestrict

/-
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/ItoIncrementMomentRestrict.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
universe u
variable {Ω : Type u} [MeasurableSpace Ω]

Formal statement


theorem LevyStochCalc.Brownian.Ito.SimplePredictable.varClock_le_Ioc_readable {T : ℝ} (H : SimplePredictable Ω T) {a b C : ℝ}
    (hab : a ≤ b) :
    H.varClock (fun j => if a ≤ H.partition j.castSucc ∧ H.partition j.succ ≤ b then C else 0)
        H.N ≤ C ^ 2 * (b - a) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoIncrementMomentRestrict.lean#L75-L134 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoIncrementMomentRestrict`, declaration `LevyStochCalc.Brownian.Ito.SimplePredictable.varClock_le_Ioc`; 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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