Second Moment Bound for a Vector Itô-Process Increment

Proved

LevyStochCalc.Brownian.Ito.integral_sq_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 second-moment estimate for the norm of a vector Itô-process increment.

Let $(\Omega,\mathcal F,P)$ be a probability space, let $W$ be a $d$ -dimensional Brownian motion with coordinate Brownian filtrations, and let $H_{mk}$ be measurable, progressively measurable integrands whose squared norms have finite extended integrals on every positive interval. Assume $C\ge0$ and $|H_{mk}(\omega,s)|\le C$ . Let $X_0:\Omega\to\operatorname{Fin}(n)\to\mathbb R$ and let $b_m(\omega,s)$ be measurable drift coefficients. Assume that $B\ge0$ and $|b_m(\omega,s)|\le B$ for every $m,\omega,s$ . For $0\le u<v$ , consider vectorItoProcess built from these data.

The increment is square-integrable and

\int_\Omega \left\|X_v(\omega)-X_u(\omega)\right\|^2\,dP(\omega) \le n\left(2\bigl(B(v-u)\bigr)^2+2d^2C^2(v-u)\right).

This provides the coordinate-to-vector second-moment estimate used in later window and grid bounds.

Formalization Note The norm is the norm on $\operatorname{Fin}(n)\to\mathbb R$ , and all natural-number factors are coerced to $\mathbb R$ . The conclusion also asserts integrability of the squared norm.

Original Lean declaration header

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

theorem integral_sq_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‖ ^ 2) 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‖ ^ 2 ∂P
        ≤ (n : ℝ) * (2 * (B * (v - u)) ^ 2 + 2 * ((d : ℝ) ^ 2 * (C ^ 2 * (v - u))))

Module navigation

Module: LevyStochCalc/Brownian/VectorItoProcess.

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

Supporting theorems: integrable_abs_drift_pow, integrable_sq_add_and_le, integral_sq_vectorItoMartingale_sub_le, measurable_vectorItoProcess_sub, abs_setIntegral_Ioc_le.

Previous: integral_norm_vectorItoProcess_sub_le · Next: integrable_norm_vectorItoProcess_sub_pow_three.

Source excerpt, lines 494–499 · 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_sq_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‖ ^ 2) 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‖ ^ 2 ∂P
        ≤ (n : ℝ) * (2 * (B * (v - u)) ^ 2 + 2 * ((d : ℝ) ^ 2 * (C ^ 2 * (v - u)))) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoProcess.lean#L492-L539 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoProcess`, declaration `LevyStochCalc.Brownian.Ito.integral_sq_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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