First Absolute Moment of a Scalar Itô Increment

Proved

LevyStochCalc.Brownian.Ito.integral_abs_sub_stochasticIntegral_le_readable

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

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

This is a first-absolute-moment estimate for an increment of a scalar Itô integral.

Let $(\Omega,P)$ be a probability space, let $W$ have Brownian filtration $\mathcal F$ , and let $H$ be measurable and progressively measurable. Assume the integrated squared norm of $H$ is finite on every positive time interval, $C\ge0$ , and $|H(\omega,s)|\le C$ for all $\omega,s$ . For $0\le u<v$ , set $M_t=\texttt{stochasticIntegralBrownian}(W,\mathcal F,H,t)$ .

Then $M_v-M_u$ is integrable in absolute value and

\int |M_v-M_u|\,dP\le C\sqrt{v-u}.

The estimate is the scalar martingale-increment bound used in subsequent vector and Itô-process estimates.

Original Lean declaration header

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

theorem integral_abs_sub_stochasticIntegral_le {u v : ℝ} (hu : 0 ≤ u) (huv : u < v) :
    MeasureTheory.Integrable (fun ω : Ω =>
        |stochasticIntegralBrownian W ℱ hℱ H hm hp hq v ω
          - stochasticIntegralBrownian W ℱ hℱ H hm hp hq u ω|) P
      ∧ ∫ ω, |stochasticIntegralBrownian W ℱ hℱ H hm hp hq v ω
            - stochasticIntegralBrownian W ℱ hℱ H hm hp hq u ω| ∂P
          ≤ C * Real.sqrt (v - u)

Module navigation

Module: LevyStochCalc/Brownian/ItoProcess.

Definition modules: LevyStochCalc/Brownian/ItoL2CompletionMaster, LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Construction, LevyStochCalc/Probability/Progressive.

Supporting theorems: integral_sub_sq_le, integral_abs_le_sqrt_of_integral_sq_le.

Previous: integrable_abs_drift_pow.

Source excerpt, lines 263–269 · 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 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 < ⊤)
variable {C : ℝ} (hC0 : 0 ≤ C) (hCH : ∀ ω s, |H ω s| ≤ C)
include hℱ hC0 hCH

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_abs_sub_stochasticIntegral_le_readable {u v : ℝ} (hu : 0 ≤ u) (huv : u < v) :
    MeasureTheory.Integrable (fun ω : Ω =>
        |stochasticIntegralBrownian W ℱ hℱ H hm hp hq v ω
          - stochasticIntegralBrownian W ℱ hℱ H hm hp hq u ω|) P
      ∧ ∫ ω, |stochasticIntegralBrownian W ℱ hℱ H hm hp hq v ω
            - stochasticIntegralBrownian W ℱ hℱ H hm hp hq u ω| ∂P
          ≤ C * Real.sqrt (v - u) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoProcess.lean#L261-L286 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integral_abs_sub_stochasticIntegral_le`; 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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