Evaluation of a Window Step Integrand

Proved

LevyStochCalc.Brownian.Ito.stepIocMul_eval_readable

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

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

This is an explicit evaluation formula for the simple predictable process stepIocMul.

Let $(\Omega,\mathcal A)$ be a measurable space, let $0<a<b$ , and let $V:\Omega\to\mathbb R$ be bounded and measurable. For every real time $s$ and sample point $\omega$ , the evaluation of stepIocMul Ω ha hab V hVb hVm equals $V(\omega)$ multiplied by the indicator of the interval $(a,b]$ :

\operatorname{stepIocMul}(V)(s,\omega) =V(\omega)\,\mathbf 1_{(a,b]}(s).

The identity is useful for replacing a simple-process representation by the corresponding pointwise window formula.

Original Lean declaration header

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

theorem stepIocMul_eval {a b : ℝ} (ha : 0 < a) (hab : a < b) {V : Ω → ℝ}
    (hVb : ∃ M : ℝ, ∀ ω, |V ω| ≤ M) (hVm : Measurable V) (s : ℝ) (ω : Ω) :
    (stepIocMul Ω ha hab V hVb hVm).eval s ω = V ω * indIoc Ω a b ω s

Module navigation

Module: LevyStochCalc/Brownian/ItoPullOut.

Definition modules: LevyStochCalc/Brownian/ItoSimpleIntegrand, LevyStochCalc/Brownian/ItoIncrement.

Next: stepIocMul₀_eval.

Source excerpt, lines 68–70 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoPullOut

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Modified 2026-10-12 by the KLS formalization project: ported to Lean v4.33.0 and
-- Mathlib db584cd. The changes are listed in vendor/README.md and vendor/port.patch.
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/ItoPullOut.lean; Apache-2.0; locally patched, see formalization/vendor/port.patch; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
universe u
variable {Ω : Type u} [MeasurableSpace Ω]

Formal statement

theorem LevyStochCalc.Brownian.Ito.stepIocMul_eval_readable {a b : ℝ} (ha : 0 < a) (hab : a < b) {V : Ω → ℝ}
    (hVb : ∃ M : ℝ, ∀ ω, |V ω| ≤ M) (hVm : Measurable V) (s : ℝ) (ω : Ω) :
    (stepIocMul Ω ha hab V hVb hVm).eval s ω = V ω * indIoc Ω a b ω s := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoPullOut.lean (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoPullOut`, declaration `LevyStochCalc.Brownian.Ito.stepIocMul_eval`; the repository is private at upload time); vendored copy patched by formalization/vendor/port.patch (project file formalization/vendor/LevyStochCalc/LevyStochCalc/Brownian/ItoPullOut.lean, lines 68-85). 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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