Positive zero-based q-scale difference quotient with an endpoint convolution bound

Proved

AvramDividend.Classical.unbounded_scale_discount_zero_secant_positive_bounded

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

avram-classicaldiscountpositive-secantscale-functionzero-discount

For q>0 and a larger discount Q, the difference quotient (W^q(x)-W^0(x))/q is strictly positive and bounded by the self-convolution of W^Q for every x>0. This provides a positive, uniformly bounded family of forward secants approaching zero discount.

Preamble

import Theorems.Thm_AvramDividend_Classical_unbounded_scale_discount_uniform_secant_bound
import Theorems.Thm_AvramDividend_Classical_unbounded_scale_discount_strict_monotone
open AvramDividend.Classical MeasureTheory Set
open scoped NNReal ENNReal

Formal statement

theorem AvramDividend.Classical.unbounded_scale_discount_zero_secant_positive_bounded
    {Ω : Type*} [mΩ : MeasurableSpace Ω]
    {P : Measure Ω} {𝓕 : Filtration ℝ≥0 mΩ}
    (X : SpectrallyNegativeLevy P 𝓕) (hnbv : ¬ X.BoundedVariation)
    (q Q : ℝ) (hq : 0 < q) (hqQ : q ≤ Q)
    (W₀ Wq WQ : ℝ → ℝ)
    (hW₀ : IsScaleFunction X 0 W₀)
    (hWq : IsScaleFunction X q Wq)
    (hWQ : IsScaleFunction X Q WQ) :
    ∀ x : ℝ, 0 < x →
      0 < (Wq x - W₀ x) / q ∧
      (Wq x - W₀ x) / q ≤
        ∫ s in Ioc (0 : ℝ) x, WQ s * WQ (x - s) := by sorry

Source

Proved strict monotonicity in discount combined with the proved endpoint convolution secant bound.

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