Vector Itô process and coordinate integrals

Definition

KLSModular_LevyStochCalc_Brownian_VectorItoProcess

by Lazyfill · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitiesklskls-conjecturekls-modularlog-concavemodule-contextvendored-levystochcalc

Vendored from the LevyStochCalc library at https://github.com/cgarryZA/LevyStochCalc, commit 984dbf0b790031eac05d56fe5a84b65e15e1e932. For a $d$ -dimensional Brownian motion with coordinates satisfying IsBrownianFiltration, a coefficient array $H_{m,k}(\omega,s)$ , and the stated measurability, progressive-measurability, and finite $L^2$ -energy assumptions, coordItoIntegral is the Itô integral of $H_{m,k}$ against coordinate $k$ , and vectorItoMartingale sums these integrals over $k<d$ . vectorItoProcess is the $\mathrm{Fin}\,n\to\mathbb R$ -valued process $X_0+\int_{[0,t]}b_s\,ds+\sum_k\int_0^tH^{\cdot,k}_s\,dW^k_s$ .

Original module imports

import LevyStochCalc.Brownian.ItoProcess
import LevyStochCalc.Brownian.MultidimFiltered
import Mathlib.Algebra.Order.Chebyshev

Original declaration excerpt

Shown in the original module context.

noncomputable def coordItoIntegral (m : Fin n) (k : Fin d) (t : ℝ) (ω : Ω) : ℝ :=
  stochasticIntegralBrownian (W.W k) ℱ (hcoord k) (H m k) (hHm m k) (hHp m k) (hHs m k) t ω

Module navigation

Direct project imports: LevyStochCalc/Brownian/ItoProcess, LevyStochCalc/Brownian/MultidimFiltered.

Theorems in source order:

Project result: The KLS conjecture.

Original module · LevyStochCalc.Brownian.VectorItoProcess · Frozen source 408915f.

Definition code

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoProcess
import Definitions.Def_KLSModular_LevyStochCalc_Brownian_MultidimFiltered
import Mathlib.Algebra.Order.Chebyshev

/-
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/VectorItoProcess.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
namespace LevyStochCalc
namespace Brownian
namespace Ito
open MeasureTheory ProbabilityTheory
open scoped NNReal ENNReal
universe u
section VectorProcess
open LevyStochCalc.Brownian.Multidim
variable {Ω : Type u} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P]
  {n d : ℕ} (W : Multidim.MultidimBrownianMotion P d)
  (ℱ : Filtration ℝ ‹MeasurableSpace Ω›)
  (hcoord : ∀ k : Fin d, IsBrownianFiltration (W.W k) ℱ)
  (H : Fin n → Fin d → Ω → ℝ → ℝ)
  (hHm : ∀ m k, Measurable (Function.uncurry (H m k)))
  (hHp : ∀ m k, Probability.ProgressivelyMeasurable ℱ (H m k))
  (hHs : ∀ (m : Fin n) (k : Fin d) (T : ℝ), 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
    (‖H m k ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
/-- The Itô integral of the `(m, k)` entry of the diffusion matrix against the `k`-th Brownian
coordinate. -/
noncomputable def coordItoIntegral (m : Fin n) (k : Fin d) (t : ℝ) (ω : Ω) : ℝ :=
  stochasticIntegralBrownian (W.W k) ℱ (hcoord k) (H m k) (hHm m k) (hHp m k) (hHs m k) t ω
/-- The martingale part of the `m`-th coordinate of a vector Itô process. -/
noncomputable def vectorItoMartingale (m : Fin n) (t : ℝ) (ω : Ω) : ℝ :=
  ∑ k : Fin d, coordItoIntegral W ℱ hcoord H hHm hHp hHs m k t ω
/-- The vector Itô process `X_t = X₀ + ∫_{[0,t]} b_s ds + ∑ₖ ∫_0^t H^{·,k}_s dWᵏ_s`. -/
noncomputable def vectorItoProcess (X₀ : Ω → Fin n → ℝ) (bdrift : Fin n → Ω → ℝ → ℝ)
    (t : ℝ) (ω : Ω) : Fin n → ℝ := fun m =>
  X₀ ω m + (∫ s in Set.Icc (0 : ℝ) t, bdrift m ω s ∂volume)
    + vectorItoMartingale W ℱ hcoord H hHm hHp hHs m t ω
variable {C : ℝ} (hC0 : 0 ≤ C)
  (hCH : ∀ (m : Fin n) (k : Fin d) (ω : Ω) (s : ℝ), |H m k ω s| ≤ C)
end VectorProcess
end Ito
end Brownian
end LevyStochCalc

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/VectorItoProcess.lean#L217-L230 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.VectorItoProcess`, definition bundle `KLSModular_LevyStochCalc_Brownian_VectorItoProcess`; 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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