Positive causal convolution of two discount scale functions

Proved

AvramDividend.Classical.unbounded_scale_discount_convolution_positive

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

avram-classicalconvolutiondiscountpositivityscale-function

For any ordered distinct nonnegative discount rates of an unbounded-variation spectrally negative Lévy process, the causal convolution of their two scale functions is strictly positive at every positive reserve. The proof uses the proved exact discount finite-difference formula and strict discount monotonicity.

Preamble

import Theorems.Thm_AvramDividend_Classical_unbounded_scale_discount_secant_convolution
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_convolution_positive
    {Ω : Type*} [mΩ : MeasurableSpace Ω]
    {P : Measure Ω} {𝓕 : Filtration ℝ≥0 mΩ}
    (X : SpectrallyNegativeLevy P 𝓕) (hnbv : ¬ X.BoundedVariation)
    (q₁ q₂ : ℝ) (hq₁ : 0 ≤ q₁) (hq₂ : 0 ≤ q₂) (hq : q₁ < q₂)
    (W₁ W₂ : ℝ → ℝ)
    (hW₁ : IsScaleFunction X q₁ W₁) (hW₂ : IsScaleFunction X q₂ W₂) :
    ∀ x : ℝ, 0 < x →
      0 < ∫ s in Ioc (0 : ℝ) x, W₁ s * W₂ (x - s) := by sorry

Source

Corollary of the Proved discount secant-convolution identity and strict monotonicity.

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