Integrability of Powers of a Bounded Drift Increment

Proved

LevyStochCalc.Brownian.Ito.integrable_abs_drift_pow_readable

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

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

This proves integrability of powers of a bounded drift integral.

Let $(\Omega,\mathcal A,P)$ be a probability space and let $b^{\mathrm{drift}}:\Omega\times\mathbb R\to\mathbb R$ be jointly measurable. Assume that for some $B\ge0$ ,

|b^{\mathrm{drift}}(\omega,s)|\le B

for every $\omega,s$ . If $u ot gtr v$ is not assumed; rather, assume $u\le v$ . Then for every $n\in\mathbb N$ ,

\omega\longmapsto \left|\int_{(u,v]}b^{\mathrm{drift}}(\omega,s)\,ds\right|^n

is integrable with respect to $P$ .

This supplies the all-orders integrability of bounded drift increments.

Original Lean declaration header

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

theorem integrable_abs_drift_pow
    (bdrift : Ω → ℝ → ℝ) (hbm : Measurable (Function.uncurry bdrift))
    {B : ℝ} (hB0 : 0 ≤ B) (hB : ∀ (ω : Ω) (s : ℝ), |bdrift ω s| ≤ B)
    {u v : ℝ} (huv : u ≤ v) (n : ℕ) :
    MeasureTheory.Integrable
      (fun ω : Ω => |∫ s in Set.Ioc u v, bdrift ω s ∂volume| ^ n) P

Module navigation

Module: LevyStochCalc/Brownian/ItoProcess.

Supporting theorems: abs_setIntegral_Ioc_le.

Next: integral_abs_sub_stochasticIntegral_le.

Source excerpt, lines 49–54 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoProcess

/-
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/ItoProcess.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
universe u
variable {Ω : Type u} [MeasurableSpace Ω]
variable {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.integrable_abs_drift_pow_readable
    (bdrift : Ω → ℝ → ℝ) (hbm : Measurable (Function.uncurry bdrift))
    {B : ℝ} (hB0 : 0 ≤ B) (hB : ∀ (ω : Ω) (s : ℝ), |bdrift ω s| ≤ B)
    {u v : ℝ} (huv : u ≤ v) (n : ℕ) :
    MeasureTheory.Integrable
      (fun ω : Ω => |∫ s in Set.Ioc u v, bdrift ω s ∂volume| ^ n) P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoProcess.lean#L48-L65 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integrable_abs_drift_pow`; 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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