L² Membership of a Squared Itô Increment

Proved

LevyStochCalc.Brownian.Ito.memLp_two_sq_increment_readable

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

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

Let $W$ be a Brownian motion with Brownian filtration $\mathcal F$ , and let $H$ be measurable, progressively measurable, and jointly square-integrable on every positive finite horizon. Suppose $C\ge0$ and $|H(\omega,s)|\le C$ for all $\omega,s$ . If $0\le a<b$ , then

\omega\mapsto\bigl(F_b(\omega)-F_a(\omega)\bigr)^2\in L^2(P),

where $F_t=\mathrm{stochasticIntegralBrownian}(W,\mathcal F,H,t)$ . Equivalently, the displayed squared increment has MemLp exponent $2$ under $P$ .

Original Lean declaration header

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

theorem memLp_two_sq_increment {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    MemLp (fun ω => (stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
      - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω) ^ 2) 2 P

Module navigation

Module: LevyStochCalc/Brownian/ItoQuadVarSum.

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

Supporting theorems: lintegral_stochasticIntegralBrownian_sub_pow_four_le.

Previous: compensator_sub_bounds · Next: integral_sq_quadVarIncrement_le.

Source excerpt, lines 225–227 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoQuadVarSum

/-
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/ItoQuadVarSum.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]
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) (hm : Measurable (Function.uncurry H))
  (hp : LevyStochCalc.Probability.ProgressivelyMeasurable ℱ H)
  (hq : ∀ T, 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H ω s‖₊ : ℝ≥0∞) ^ 2 ∂MeasureTheory.MeasureSpace.volume ∂P < ⊤)
  {C : ℝ} (hC0 : 0 ≤ C) (hCH : ∀ ω s, |H ω s| ≤ C)
include hℱ hC0 hCH

Formal statement


theorem LevyStochCalc.Brownian.Ito.memLp_two_sq_increment_readable {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    MemLp (fun ω => (stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
      - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω) ^ 2) 2 P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoQuadVarSum.lean#L223-L244 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoQuadVarSum`, declaration `LevyStochCalc.Brownian.Ito.memLp_two_sq_increment`; 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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