Mixed scale-function convolutions commute at distinct discounts

Proved

AvramDividend.Classical.unbounded_scale_mixed_convolution_commutative

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

avram-classicalconvolutiondiscountresolventscale-functionsymmetry

For an unbounded-variation spectrally negative Lévy process and two distinct nonnegative discounts, the causal convolution of their q-scale functions is invariant under exchanging the factors. This follows from two orientations of the resolvent identity and cancellation of the nonzero discount gap.

Preamble

import Mathlib.Tactic.Linarith
import Theorems.Thm_AvramDividend_Classical_unbounded_scale_resolvent_convolution
open AvramDividend.Classical MeasureTheory Set
open scoped NNReal ENNReal

Formal statement

theorem AvramDividend.Classical.unbounded_scale_mixed_convolution_commutative
    {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)
    (hqne : q₁ ≠ q₂) (W₁ W₂ : ℝ → ℝ)
    (hW₁ : IsScaleFunction X q₁ W₁)
    (hW₂ : IsScaleFunction X q₂ W₂) :
    ∀ x : ℝ, 0 < x →
      (∫ s in Ioc (0 : ℝ) x, W₁ s * W₂ (x - s)) =
        ∫ s in Ioc (0 : ℝ) x, W₂ s * W₁ (x - s) := by sorry

Source

Algebraic corollary of the proved unconditional q-scale resolvent convolution theorem.

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