First Norm Moment of a Vector Itô-Process Increment

Proved

LevyStochCalc.Brownian.Ito.integral_norm_vectorItoProcess_sub_le_readable

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

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

This is a first-moment bound for the supremum norm of a vector Itô-process increment.

Let $P$ be a probability measure, let $W$ be a $d$ -dimensional Brownian motion with coordinate Brownian filtrations, and let the integrands $H_{mk}$ satisfy joint measurability, progressive measurability, finite $\mathrm{ENNReal}$ squared integrals on positive intervals, and $|H_{mk}|\le C$ with $C\ge0$ . Let $X=\texttt{vectorItoProcess}(W,\mathcal F,H,X_0,b)$ , where each drift coordinate is measurable, $B\ge0$ , and $|b_m(\omega,s)|\le B$ . For $0\le u<v$ ,

\int \|X_v-X_u\|\,dP \le n\left(B(v-u)+dC\sqrt{v-u}\right),

where the norm is the norm on $\mathrm{Fin}(n)\to\mathbb R$ .

The norm increment is also integrable. This estimate combines coordinatewise drift and martingale bounds for later window and Riemann-sum estimates.

Original Lean declaration header

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

theorem integral_norm_vectorItoProcess_sub_le {u v : ℝ} (hu : 0 ≤ u) (huv : u < v) :
    Integrable (fun ω => ‖vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift v ω
        - vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift u ω‖) P
      ∧ ∫ ω, ‖vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift v ω
          - vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift u ω‖ ∂P
        ≤ (n : ℝ) * (B * (v - u) + (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_add_and_le, integrable_abs_drift_pow, integral_abs_vectorItoMartingale_sub_le, measurable_vectorItoProcess_sub, abs_setIntegral_Ioc_le.

Previous: measurable_vectorItoProcess_sub · Next: integral_sq_norm_vectorItoProcess_sub_le.

Source excerpt, lines 446–451 · 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)
variable (X₀ : Ω → Fin n → ℝ) (bdrift : Fin n → Ω → ℝ → ℝ)
  (hbm : ∀ m, Measurable (Function.uncurry (bdrift m))) {B : ℝ} (hB0 : 0 ≤ B)
  (hB : ∀ (m : Fin n) (ω : Ω) (s : ℝ), |bdrift m ω s| ≤ B)
include hbm hC0 hCH hB0 hB

Formal statement


theorem LevyStochCalc.Brownian.Ito.integral_norm_vectorItoProcess_sub_le_readable {u v : ℝ} (hu : 0 ≤ u) (huv : u < v) :
    Integrable (fun ω => ‖vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift v ω
        - vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift u ω‖) P
      ∧ ∫ ω, ‖vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift v ω
          - vectorItoProcess W ℱ hcoord H hHm hHp hHs X₀ bdrift u ω‖ ∂P
        ≤ (n : ℝ) * (B * (v - u) + (d : ℝ) * (C * Real.sqrt (v - u))) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoProcess.lean#L444-L490 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integral_norm_vectorItoProcess_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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