Bounds for a Compensator Increment

Proved

LevyStochCalc.Brownian.Ito.compensator_sub_bounds_readable

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

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

This theorem bounds the increment of the quadratic compensator of a bounded integrand.

Let $H:\Omega\times\mathbb R\to\mathbb R$ be jointly measurable, let $C\ge0$ , and assume $|H(\omega,s)|\le C$ . For $0\le a\le b$ and every $\omega\in\Omega$ ,

0\le \int_{[0,b]}H(\omega,u)^2\,du-\int_{[0,a]}H(\omega,u)^2\,du \le C^2(b-a).

The result identifies the compensator increment over $(a,b]$ with a nonnegative integral and bounds it using the interval length. It is a deterministic estimate for compensated quadratic-variation increments.

Original Lean declaration header

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

theorem compensator_sub_bounds {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) (ω : Ω) :
    0 ≤ (∫ u in Set.Icc (0 : ℝ) b, (H ω u) ^ 2 ∂volume)
          - ∫ u in Set.Icc (0 : ℝ) a, (H ω u) ^ 2 ∂volume
      ∧ (∫ u in Set.Icc (0 : ℝ) b, (H ω u) ^ 2 ∂volume)
          - ∫ u in Set.Icc (0 : ℝ) a, (H ω u) ^ 2 ∂volume ≤ C ^ 2 * (b - a)

Module navigation

Module: LevyStochCalc/Brownian/ItoQuadVarSum.

Previous: condExp_quadVarIncrement · Next: memLp_two_sq_increment.

Source excerpt, lines 178–182 · 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 < ⊤)
  {C : ℝ} (hC0 : 0 ≤ C) (hCH : ∀ ω s, |H ω s| ≤ C)
include hm hC0 hCH

Formal statement


theorem LevyStochCalc.Brownian.Ito.compensator_sub_bounds_readable {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) (ω : Ω) :
    0 ≤ (∫ u in Set.Icc (0 : ℝ) b, (H ω u) ^ 2 ∂volume)
          - ∫ u in Set.Icc (0 : ℝ) a, (H ω u) ^ 2 ∂volume
      ∧ (∫ u in Set.Icc (0 : ℝ) b, (H ω u) ^ 2 ∂volume)
          - ∫ u in Set.Icc (0 : ℝ) a, (H ω u) ^ 2 ∂volume ≤ C ^ 2 * (b - a) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoQuadVarSum.lean#L176-L208 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoQuadVarSum`, declaration `LevyStochCalc.Brownian.Ito.compensator_sub_bounds`; 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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