Coordinate Decomposition of the Second-Order Taylor Sum

Proved

LevyStochCalc.Brownian.Ito.sum_apply_fderiv_two_eq_readable

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

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

This is the coordinate expansion of a second-order Taylor sum.

Let $n,m\in\mathbb N$ , let $Y_i\in\operatorname{Fin}(n)\to\mathbb R$ be a sequence, and let $f''$ assign to each state a continuous bilinear map. Then

\sum_{i<m}\frac{f''(Y_i)(Y_{i+1}-Y_i)(Y_{i+1}-Y_i)}{2} =\frac12\sum_{p<n}\sum_{q<n}\sum_{i<m} \partial_{pq}f''(Y_i) \bigl(Y_{i+1,p}-Y_{i,p}\bigr)\bigl(Y_{i+1,q}-Y_{i,q}\bigr).

This is an algebraic coordinate identity used to express the quadratic Taylor term in the vector Itô formula.

Original Lean declaration header

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

theorem sum_apply_fderiv₂_eq (f'' : (Fin n → ℝ) → (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) →L[ℝ] ℝ)
    (Y : ℕ → Fin n → ℝ) (m : ℕ) :
    ∑ i ∈ Finset.range m, f'' (Y i) (Y (i + 1) - Y i) (Y (i + 1) - Y i) / 2
      = 1 / 2 * ∑ p : Fin n, ∑ q : Fin n, ∑ i ∈ Finset.range m,
          coordDeriv₂ f'' p q (Y i) * ((Y (i + 1) p - Y i p) * (Y (i + 1) q - Y i q))

Module navigation

Module: LevyStochCalc/Brownian/VectorItoFormulaDecomp.

Definition modules: LevyStochCalc/Brownian/CoordDerivative.

Previous: sum_range_add_sum_comm · Next: sum_apply_fderiv_eq.

Source excerpt, lines 80–84 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_VectorItoFormulaDecomp

/-
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/VectorItoFormulaDecomp.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
variable {n : ℕ}

Formal statement


theorem LevyStochCalc.Brownian.Ito.sum_apply_fderiv_two_eq_readable (f'' : (Fin n → ℝ) → (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) →L[ℝ] ℝ)
    (Y : ℕ → Fin n → ℝ) (m : ℕ) :
    ∑ i ∈ Finset.range m, f'' (Y i) (Y (i + 1) - Y i) (Y (i + 1) - Y i) / 2
      = 1 / 2 * ∑ p : Fin n, ∑ q : Fin n, ∑ i ∈ Finset.range m,
          coordDeriv₂ f'' p q (Y i) * ((Y (i + 1) p - Y i p) * (Y (i + 1) q - Y i q)) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoFormulaDecomp.lean#L79-L99 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoFormulaDecomp`, declaration `LevyStochCalc.Brownian.Ito.sum_apply_fderiv₂_eq`; 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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