Squared Error of a Martingale Riemann Sum

Proved

LevyStochCalc.Brownian.Ito.IsVectorItoVersion.lintegral_sq_martingaleRiemann_sub_le_readable

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

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

This is an $L^2$ error estimate for a martingale Riemann sum along a vector Itô process.

Let $(\Omega,\mathcal F,P)$ , the $d$ -dimensional Brownian motion, its coordinate filtrations, the integrands $H_{pk}$ , and the vector Itô-process version $X$ satisfy the stated measurability, progressive measurability, finite squared-integral, and IsVectorItoVersion hypotheses. Assume $C\ge0$ , $|H_{pk}|\le C$ , measurable drifts bounded by $B\ge0$ , and let $\varphi$ satisfy $|\varphi(x)|\le K_\varphi$ and

|\varphi(x)-\varphi(y)|\le L\|x-y\|, \qquad L\ge0.

Fix $k\in\operatorname{Fin}(d)$ . Let $G$ be measurable and progressively measurable with finite squared extended integrals and $|G(\omega,s)|\le C_G$ for $C_G\ge0$ . Assume also that $\varphi(X_s)G_s$ is measurable, progressively measurable, and has finite squared extended integrals. For $T>0$ and $m\ne0$ , the frozen-weight sum satisfies

\int_\Omega\left|\sum_{i<m}\varphi(X_{t_i})\bigl(I_G(t_{i+1})-I_G(t_i)\bigr)-I_{\varphi(X)G}(T)\right|^2dP \le \operatorname{ofReal}(C_G^2)\operatorname{ofReal}(L^2) \operatorname{ofReal}\!\left(Tn\left(2\left(B\frac{T}{m}\right)^2+2d^2C^2\frac{T}{m}\right)\right),

where $t_i=\operatorname{unifGrid}(T,m,i)$ .

This is the quantitative martingale Riemann-sum approximation used in the vector Itô formula.

Formalization Note The squared norm integral is an ENNReal integral. The boundedness of $\varphi$ is used as a hypothesis even though the conclusion is the displayed error bound.

Original Lean declaration header

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

theorem IsVectorItoVersion.lintegral_sq_martingaleRiemann_sub_le
    (h : IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X)
    (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)
    {φ : (Fin n → ℝ) → ℝ} {Kφ : ℝ} (hφbd : ∀ x, |φ x| ≤ Kφ)
    {L : ℝ} (hL0 : 0 ≤ L) (hφlip : ∀ x y : Fin n → ℝ, |φ x - φ y| ≤ L * ‖x - y‖)
    (k : Fin d) {G : Ω → ℝ → ℝ} (hGm : Measurable (Function.uncurry G))
    (hGp : Probability.ProgressivelyMeasurable ℱ G)
    (hGs : ∀ T', 0 < T' → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T',
      (‖G ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {CG : ℝ} (hCG0 : 0 ≤ CG) (hCG : ∀ ω s, |G ω s| ≤ CG)
    (hmg : Measurable (Function.uncurry fun ω s => φ (X s ω) * G ω s))
    (hpg : Probability.ProgressivelyMeasurable ℱ fun ω s => φ (X s ω) * G ω s)
    (hqg : ∀ T', 0 < T' → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T',
      (‖φ (X s ω) * G ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    ∫⁻ ω, (‖(∑ i : Fin m, φ (X (unifGrid T m (i : ℕ)) ω)
          * (stochasticIntegralBrownian (W.W k) ℱ (hcoord k) G hGm hGp hGs
              (unifGrid T m ((i : ℕ) + 1)) ω
            - stochasticIntegralBrownian (W.W k) ℱ (hcoord k) G hGm hGp hGs
              (unifGrid T m (i : ℕ)) ω))
        - stochasticIntegralBrownian (W.W k) ℱ (hcoord k) (fun ω s => φ (X s ω) * G ω s)
            hmg hpg hqg T ω‖₊ : ℝ≥0∞) ^ 2 ∂P
      ≤ ENNReal.ofReal (CG ^ 2) * (ENNReal.ofReal (L ^ 2)
        * ENNReal.ofReal (T * ((n : ℝ) * (2 * (B * (T / (m : ℝ))) ^ 2
          + 2 * ((d : ℝ) ^ 2 * (C ^ 2 * (T / (m : ℝ))))))))

Module navigation

Module: LevyStochCalc/Brownian/VectorItoMartingaleRiemann.

Definition modules: LevyStochCalc/Brownian/ItoL2CompletionMaster, LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Multidim, LevyStochCalc/Brownian/ItoFormulaGrid, LevyStochCalc/Probability/Progressive, LevyStochCalc/Brownian/VectorItoProcessVersion.

Supporting theorems: IsVectorItoVersion.lintegral_sq_frozen_sub_le, lintegral_sq_stochasticIntegral_mul_sub_le, stochasticIntegralBrownian_integralAgainst.

Previous: IsVectorItoVersion.lintegral_sq_frozen_sub_le.

Source excerpt, lines 161–186 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoMartingaleRiemann

/-
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/VectorItoMartingaleRiemann.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 < ⊤}
  {X₀ : Ω → Fin n → ℝ} {bdrift : Fin n → Ω → ℝ → ℝ} {X : ℝ → Ω → Fin n → ℝ}
  {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.IsVectorItoVersion.lintegral_sq_martingaleRiemann_sub_le_readable
    (h : IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X)
    (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)
    {φ : (Fin n → ℝ) → ℝ} {Kφ : ℝ} (hφbd : ∀ x, |φ x| ≤ Kφ)
    {L : ℝ} (hL0 : 0 ≤ L) (hφlip : ∀ x y : Fin n → ℝ, |φ x - φ y| ≤ L * ‖x - y‖)
    (k : Fin d) {G : Ω → ℝ → ℝ} (hGm : Measurable (Function.uncurry G))
    (hGp : Probability.ProgressivelyMeasurable ℱ G)
    (hGs : ∀ T', 0 < T' → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T',
      (‖G ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {CG : ℝ} (hCG0 : 0 ≤ CG) (hCG : ∀ ω s, |G ω s| ≤ CG)
    (hmg : Measurable (Function.uncurry fun ω s => φ (X s ω) * G ω s))
    (hpg : Probability.ProgressivelyMeasurable ℱ fun ω s => φ (X s ω) * G ω s)
    (hqg : ∀ T', 0 < T' → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T',
      (‖φ (X s ω) * G ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    {T : ℝ} (hT : 0 < T) {m : ℕ} (hm0 : m ≠ 0) :
    ∫⁻ ω, (‖(∑ i : Fin m, φ (X (unifGrid T m (i : ℕ)) ω)
          * (stochasticIntegralBrownian (W.W k) ℱ (hcoord k) G hGm hGp hGs
              (unifGrid T m ((i : ℕ) + 1)) ω
            - stochasticIntegralBrownian (W.W k) ℱ (hcoord k) G hGm hGp hGs
              (unifGrid T m (i : ℕ)) ω))
        - stochasticIntegralBrownian (W.W k) ℱ (hcoord k) (fun ω s => φ (X s ω) * G ω s)
            hmg hpg hqg T ω‖₊ : ℝ≥0∞) ^ 2 ∂P
      ≤ ENNReal.ofReal (CG ^ 2) * (ENNReal.ofReal (L ^ 2)
        * ENNReal.ofReal (T * ((n : ℝ) * (2 * (B * (T / (m : ℝ))) ^ 2
          + 2 * ((d : ℝ) ^ 2 * (C ^ 2 * (T / (m : ℝ)))))))) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoMartingaleRiemann.lean#L156-L228 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoMartingaleRiemann`, declaration `LevyStochCalc.Brownian.Ito.IsVectorItoVersion.lintegral_sq_martingaleRiemann_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).

View graph