Second Moment Bound for a Vector Itô Martingale Increment

This is a technical second-moment estimate for a coordinate of a vector Itô martingale.

Let $(\Omega,\mathcal F,P)$ be a probability space, let $W$ be a $d$-dimensional Brownian motion whose coordinates have Brownian filtration $\mathcal F$, and let $H_{mk}(\omega,s)$ be the integrands. Assume that every $H_{mk}$ is measurable and progressively measurable, that its squared norm has finite extended integral on every interval $[0,T]$ with $T>0$, and that there is $C\ge 0$ such that

$$|H_{mk}(\omega,s)|\le C$$

for all state indices $m\in\operatorname{Fin}(n)$, Brownian indices $k\in\operatorname{Fin}(d)$, $\omega\in\Omega$, and times $s\in\mathbb R$. For $u,v\in\mathbb R$ with $0\le u<v$, the theorem concerns the coordinate $m$ of vectorItoMartingale.

The increment is square-integrable and satisfies

$$\int_\Omega \left|M_m(v,\omega)-M_m(u,\omega)\right|^2\,dP(\omega) \le d^2 C^2(v-u),$$

where $M_m(t,\omega)$ denotes the corresponding coordinate of vectorItoMartingale.

This estimate is reused as the coordinatewise martingale moment bound for vector Itô-process estimates.

Formalization Note P is a probability measure. The finiteness hypothesis is expressed using an ENNReal integral, while the conclusion uses the ordinary real integral; vectorItoMartingale is the Lean definition involved.

Original Lean declaration header

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

theorem integral_sq_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) ^ 2) P
      ∧ ∫ ω, (vectorItoMartingale W ℱ hcoord H hHm hHp hHs m v ω
          - vectorItoMartingale W ℱ hcoord H hHm hHp hHs m u ω) ^ 2 ∂P
        ≤ (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_sq_sum_and_le, integral_sub_sq_le.

Previous: integrable_abs_add_and_le · Next: integral_abs_vectorItoMartingale_sub_le.

Source excerpt, lines 285–290 · Frozen source 408915f.

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

theorem LevyStochCalc.Brownian.Ito.integral_sq_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) ^ 2) P
      ∧ ∫ ω, (vectorItoMartingale W ℱ hcoord H hHm hHp hHs m v ω
          - vectorItoMartingale W ℱ hcoord H hHm hHp hHs m u ω) ^ 2 ∂P
        ≤ (d : ℝ) ^ 2 * (C ^ 2 * (v - u)) := by sorry