Compensated Square Martingale for the Itô Integral

Proved

LevyStochCalc.Brownian.Ito.martingale_quadVar_stochasticIntegralBrownian_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, $W$ a Brownian motion, and $\mathcal F$ a Brownian filtration. Let $H:\Omega\times\mathbb R\to\mathbb R$ be measurable and progressively measurable, and assume that for every $T>0$ the joint $L^2$ integral of $H$ over $[0,T]$ is finite. Then the process

t\longmapsto \mathrm{stochasticIntegralBrownian}(W,\mathcal F,H,t)^2- \int_{[0,t]}H(\cdot,u)^2\,du

is a martingale with respect to $\mathcal F$ and $P$ . This is the general-integrand counterpart of the simple-level compensated-square martingale.

Original Lean declaration header

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

lemma martingale_quadVar_stochasticIntegralBrownian :
    MeasureTheory.Martingale
      (fun t ω => (stochasticIntegralBrownian W ℱ hℱ H h_meas h_progMeas h_sq_int_global t ω) ^ 2
        - ∫ u in Set.Icc (0 : ℝ) t, (H ω u) ^ 2 ∂volume)
      ℱ P

Module navigation

Module: LevyStochCalc/Brownian/ItoL2CompletionQuadVar.

Definition modules: LevyStochCalc/Brownian/ItoL2CompletionMaster, LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Construction, LevyStochCalc/Probability/Progressive.

Supporting theorems: martingale_simpleIntegral_sq_sub_compensator, masterApprox_compensator_tendsto_L1, setIntegral_eval_sq_Icc_clamped, tendsto_eLpNorm_one_sq_sub.

Previous: masterApprox_compensator_tendsto_L1.

Source excerpt, lines 438–442 · 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 hℱ

Formal statement


lemma LevyStochCalc.Brownian.Ito.martingale_quadVar_stochasticIntegralBrownian_readable :
    MeasureTheory.Martingale
      (fun t ω => (stochasticIntegralBrownian W ℱ hℱ H h_meas h_progMeas h_sq_int_global t ω) ^ 2
        - ∫ u in Set.Icc (0 : ℝ) t, (H ω u) ^ 2 ∂volume)
      ℱ P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoL2CompletionQuadVar.lean#L431-L553 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoL2CompletionQuadVar`, declaration `LevyStochCalc.Brownian.Ito.martingale_quadVar_stochasticIntegralBrownian`; 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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