Master Approximation Compensator Convergence

Proved

LevyStochCalc.Brownian.Ito.masterApprox_compensator_tendsto_L1_readable

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

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

Let $P$ be a probability measure, let $\mathcal F$ be a filtration, and let $H:\Omega\times\mathbb R\to\mathbb R$ be measurable and progressively measurable. Assume that for every $T>0$ , the joint nonnegative integral of $|H|^2$ over $\Omega\times[0,T]$ is strictly below $\infty$ . For each $n$ , let $G_n=\mathrm{masterApprox}(\mathcal F,H,\ldots,n)$ be the associated simple approximation. For every $t\ge0$ ,

\left\|\omega\mapsto\int_{[0,t]}G_n.\mathrm{eval}(u,\omega)^2\,du- \int_{[0,t]}H(\omega,u)^2\,du\right\|_{L^1(P)}\longrightarrow0.

The conclusion is a convergence statement in the extended $L^1$ norm; the approximation and its hypotheses are the named Lean definitions masterApprox and ProgressivelyMeasurable.

Original Lean declaration header

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

lemma masterApprox_compensator_tendsto_L1 {t : ℝ} (ht : 0 ≤ t) :
    Filter.Tendsto (fun n => MeasureTheory.eLpNorm
      (fun ω => (∫ u in Set.Icc (0 : ℝ) t,
          ((masterApprox ℱ H h_meas h_progMeas h_sq_int_global n).eval u ω) ^ 2 ∂volume)
        - ∫ u in Set.Icc (0 : ℝ) t, (H ω u) ^ 2 ∂volume) 1 P)
      Filter.atTop (nhds 0)

Module navigation

Module: LevyStochCalc/Brownian/ItoL2CompletionQuadVar.

Definition modules: LevyStochCalc/Brownian/ItoSimpleIntegrand, LevyStochCalc/Probability/Progressive, LevyStochCalc/Brownian/ItoL2CompletionMaster.

Supporting theorems: tendsto_eLpNorm_one_sq_sub.

Next: martingale_quadVar_stochasticIntegralBrownian.

Source excerpt, lines 331–336 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoL2CompletionQuadVar

/-
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/ItoL2CompletionQuadVar.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory
open scoped NNReal ENNReal
universe u
variable {Ω : Type u} [MeasurableSpace Ω]
variable
    {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
    (W : LevyStochCalc.Brownian.BrownianMotion P)
    (ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›) (hℱ : LevyStochCalc.Brownian.IsBrownianFiltration W ℱ)
    (H : Ω → ℝ → ℝ) (h_meas : Measurable (Function.uncurry H))
    (h_progMeas : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ H)
    (h_sq_int_global : ∀ T, 0 < T →
      ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
        (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
include h_meas h_progMeas h_sq_int_global

Formal statement


lemma LevyStochCalc.Brownian.Ito.masterApprox_compensator_tendsto_L1_readable {t : ℝ} (ht : 0 ≤ t) :
    Filter.Tendsto (fun n => MeasureTheory.eLpNorm
      (fun ω => (∫ u in Set.Icc (0 : ℝ) t,
          ((masterApprox ℱ H h_meas h_progMeas h_sq_int_global n).eval u ω) ^ 2 ∂volume)
        - ∫ u in Set.Icc (0 : ℝ) t, (H ω u) ^ 2 ∂volume) 1 P)
      Filter.atTop (nhds 0) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoL2CompletionQuadVar.lean#L326-L429 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoL2CompletionQuadVar`, declaration `LevyStochCalc.Brownian.Ito.masterApprox_compensator_tendsto_L1`; 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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