First Absolute Moment of a Vector Itô Martingale Increment

Proved

LevyStochCalc.Brownian.Ito.integral_abs_vectorItoMartingale_sub_le_readable

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

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

This is a coordinatewise first-moment estimate for the martingale part of a vector Itô process.

Let $P$ be a probability measure, let $W$ be a $d$ -dimensional Brownian motion whose coordinates share a filtration $\mathcal F$ , and let $H_{mk}$ be measurable, progressively measurable, and square-integrable on every positive time interval. Assume $C\ge0$ and $|H_{mk}(\omega,s)|\le C$ for every output coordinate $m\in\mathrm{Fin}(n)$ , Brownian coordinate $k\in\mathrm{Fin}(d)$ , $\omega$ , and $s$ . For $m\in\mathrm{Fin}(n)$ and $0\le u<v$ , let $M^m_t=\texttt{vectorItoMartingale}(W,\mathcal F,H,m,t)$ .

Then $M^m_v-M^m_u$ is integrable in absolute value and

\int |M^m_v-M^m_u|\,dP\le d\,C\sqrt{v-u}.

This is a technical coordinate estimate obtained by summing the scalar bounds over the $d$ driving Brownian coordinates.

Original Lean declaration header

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

theorem integral_abs_vectorItoMartingale_sub_le (m : Fin n) {u v : ℝ} (hu : 0 ≤ u) (huv : u < v) :
    Integrable (fun ω => |vectorItoMartingale W ℱ hcoord H hHm hHp hHs m v ω
        - vectorItoMartingale W ℱ hcoord H hHm hHp hHs m u ω|) P
      ∧ ∫ ω, |vectorItoMartingale W ℱ hcoord H hHm hHp hHs m v ω
          - vectorItoMartingale W ℱ hcoord H hHm hHp hHs m u ω| ∂P
        ≤ (d : ℝ) * (C * Real.sqrt (v - u))

Module navigation

Module: LevyStochCalc/Brownian/VectorItoProcess.

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

Supporting theorems: integrable_abs_sum_and_le, integral_abs_sub_stochasticIntegral_le.

Previous: integral_sq_vectorItoMartingale_sub_le · Next: measurable_vectorItoProcess_sub.

Source excerpt, lines 328–333 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoProcess

/-
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/VectorItoProcess.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] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {n d : ℕ} (W : LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion P d)
  (ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›)
  (hcoord : ∀ k : Fin d, LevyStochCalc.Brownian.IsBrownianFiltration (W.W k) ℱ)
  (H : Fin n → Fin d → Ω → ℝ → ℝ)
  (hHm : ∀ m k, Measurable (Function.uncurry (H m k)))
  (hHp : ∀ m k, LevyStochCalc.Probability.ProgressivelyMeasurable ℱ (H m k))
  (hHs : ∀ (m : Fin n) (k : Fin d) (T : ℝ), 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H m k ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
variable {C : ℝ} (hC0 : 0 ≤ C)
  (hCH : ∀ (m : Fin n) (k : Fin d) (ω : Ω) (s : ℝ), |H m k ω s| ≤ C)
include hC0 hCH

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_abs_vectorItoMartingale_sub_le_readable (m : Fin n) {u v : ℝ} (hu : 0 ≤ u) (huv : u < v) :
    Integrable (fun ω => |vectorItoMartingale W ℱ hcoord H hHm hHp hHs m v ω
        - vectorItoMartingale W ℱ hcoord H hHm hHp hHs m u ω|) P
      ∧ ∫ ω, |vectorItoMartingale W ℱ hcoord H hHm hHp hHs m v ω
          - vectorItoMartingale W ℱ hcoord H hHm hHp hHs m u ω| ∂P
        ≤ (d : ℝ) * (C * Real.sqrt (v - u)) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoProcess.lean#L326-L354 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integral_abs_vectorItoMartingale_sub_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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