Dyadic Maximum Bound for the Picard Drift

Proved

LevyStochCalc.Ito.Picard.iSup_dyadicRunMax_drift_le_readable

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

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

This bounds the dyadic running maxima of the drift component by its initial value and total variation.

Let $\mathrm{coeffs}$ be jump-diffusion coefficients, let $X$ be a state process, let $x_0\in\mathbb R^n$ , and fix a coordinate $i<n$ . Assume $T'\ge0$ and that $s\mapsto\mu(s,X_s(\omega),i)$ is integrable on $[0,T']$ . Then, for every $\omega$ ,

\sup_m\left\|\operatorname{dyadicRunMax}\bigl(t\mapsto x_{0,i}+\int_0^t\mu(s,X_s(\omega),i)\,ds,T',m\bigr)\right\|

\le |x_{0,i}|+\left|\int_0^{T'}\|\mu(s,X_s(\omega),i)\|\,ds\right|.

The displayed quantities are represented as ENNReal coercions of nonnegative norms. This is a pathwise auxiliary estimate for the drift part of the Picard step.

Original Lean declaration header

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

theorem iSup_dyadicRunMax_drift_le
    (coeffs : LevyStochCalc.Ito.Setting.JumpDiffusionCoeffs n d E)
    (X : ℝ → Ω → (Fin n → ℝ)) (x₀ : Fin n → ℝ) (i : Fin n) {T' : ℝ} (hT' : 0 ≤ T') {ω : Ω}
    (hint : MeasureTheory.IntegrableOn (fun s => coeffs.μ s (X s ω) i)
      (Set.Icc (0 : ℝ) T') volume) :
    (⨆ m, (‖dyadicRunMax (fun t ω => picardStep_drift coeffs X x₀ t ω i) T' m ω‖₊ : ℝ≥0∞))
      ≤ (‖x₀ i‖₊ : ℝ≥0∞)
        + (‖∫ s in Set.Icc (0 : ℝ) T', ‖coeffs.μ s (X s ω) i‖‖₊ : ℝ≥0∞)

Module navigation

Module: LevyStochCalc/Ito/PicardSupL2.

Definition modules: LevyStochCalc/Ito/PicardBieleckiNorm, LevyStochCalc/Ito/Setting, LevyStochCalc/Probability/DoobContinuous.

Supporting theorems: enorm_dyadicRunMax.

Next: lintegral_sq_drift_bound_lt_top.

Source excerpt, lines 59–66 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_PicardSupL2

/-
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/PicardSupL2.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory LevyStochCalc.Probability
open scoped NNReal ENNReal
universe u v
variable {Ω : Type u} [MeasurableSpace Ω] {E : Type v} [MeasurableSpace E]
variable {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n d : ℕ}
omit [MeasurableSpace Ω] [MeasurableSpace E]

Formal statement


theorem LevyStochCalc.Ito.Picard.iSup_dyadicRunMax_drift_le_readable
    (coeffs : LevyStochCalc.Ito.Setting.JumpDiffusionCoeffs n d E)
    (X : ℝ → Ω → (Fin n → ℝ)) (x₀ : Fin n → ℝ) (i : Fin n) {T' : ℝ} (hT' : 0 ≤ T') {ω : Ω}
    (hint : MeasureTheory.IntegrableOn (fun s => coeffs.μ s (X s ω) i)
      (Set.Icc (0 : ℝ) T') volume) :
    (⨆ m, (‖dyadicRunMax (fun t ω => picardStep_drift coeffs X x₀ t ω i) T' m ω‖₊ : ℝ≥0∞))
      ≤ (‖x₀ i‖₊ : ℝ≥0∞)
        + (‖∫ s in Set.Icc (0 : ℝ) T', ‖coeffs.μ s (X s ω) i‖‖₊ : ℝ≥0∞) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/PicardSupL2.lean#L57-L90 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.PicardSupL2`, declaration `LevyStochCalc.Ito.Picard.iSup_dyadicRunMax_drift_le`; 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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