Finite Sums of Martingales

Proved

LevyStochCalc.Brownian.Ito.martingale_finsetSum_readable

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

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

This vendored theorem states that finite sums preserve the martingale property.

Let $s$ be a finite set of indices, let $f_i(t,\omega)$ be a real-valued process for each index $i$ , and suppose that every $f_i$ with $i\in s$ is a martingale with respect to the filtration $\mathcal F$ and measure $P$ . Then the process

(t,\omega)\longmapsto\sum_{i\in s}f_i(t,\omega)

is also a martingale with respect to $\mathcal F$ and $P$ .

The result is a basic closure property used when assembling finite-dimensional stochastic processes.

Original Lean declaration header

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

theorem martingale_finsetSum {ι : Type*} {ℱ : Filtration ℝ ‹MeasurableSpace Ω›}
    (s : Finset ι) (f : ι → ℝ → Ω → ℝ)
    (hf : ∀ i ∈ s, MeasureTheory.Martingale (f i) ℱ P) :
    MeasureTheory.Martingale (fun t ω => ∑ i ∈ s, f i t ω) ℱ P

Module navigation

Module: LevyStochCalc/Ito/VectorItoVersionSup.

Next: lintegral_iSup_sq_norm_version_sub_le.

Source excerpt, lines 78–81 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_VectorItoVersionSup

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

Formal statement


theorem LevyStochCalc.Brownian.Ito.martingale_finsetSum_readable {ι : Type*} {ℱ : Filtration ℝ ‹MeasurableSpace Ω›}
    (s : Finset ι) (f : ι → ℝ → Ω → ℝ)
    (hf : ∀ i ∈ s, MeasureTheory.Martingale (f i) ℱ P) :
    MeasureTheory.Martingale (fun t ω => ∑ i ∈ s, f i t ω) ℱ P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/VectorItoVersionSup.lean#L77-L97 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.VectorItoVersionSup`, declaration `LevyStochCalc.Brownian.Ito.martingale_finsetSum`; 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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