Brownian Filtration Property Under Right-Continuous Modification

Proved

LevyStochCalc.Brownian.IsBrownianFiltration.rightCont_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 on a probability space $(\Omega,P)$ and let $\mathcal F$ be a filtration. If IsBrownianFiltration W ℱ holds, then $W$ is also Brownian with respect to the right-continuous filtration ℱ.rightCont:

\operatorname{IsBrownianFiltration}(W,\mathcal F)\ \Longrightarrow\ \operatorname{IsBrownianFiltration}(W,\mathcal F_+).

The conclusion preserves the adaptedness and future-increment independence required by the Brownian-filtration predicate.

Original Lean declaration header

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

theorem IsBrownianFiltration.rightCont {W : BrownianMotion P} {ℱ : Filtration ℝ ‹MeasurableSpace Ω›}
    (h : IsBrownianFiltration W ℱ) : IsBrownianFiltration W ℱ.rightCont

Module navigation

Module: LevyStochCalc/Brownian/Filtered.

Definition modules: LevyStochCalc/Brownian/Construction.

Supporting theorems: indep_comap_of_tendsto_ae, indep_iSup_of_finset.

Source excerpt, lines 68–69 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_Filtered

/-
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/Filtered.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory Filter Topology
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.IsBrownianFiltration.rightCont_readable {W : BrownianMotion P} {ℱ : Filtration ℝ ‹MeasurableSpace Ω›}
    (h : IsBrownianFiltration W ℱ) : IsBrownianFiltration W ℱ.rightCont := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/Filtered.lean#L67-L127 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.Filtered`, declaration `LevyStochCalc.Brownian.IsBrownianFiltration.rightCont`; 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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