Zero-barrier candidate value vanishes for unbounded-variation Lévy processes

Proved

AvramDividend.Classical.unbounded_variation_zero_barrier_candidate_vanishes

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

avram-dividendboundary-valuescale-functionunbounded-variationzero-barrier

For any positive discount and unbounded-variation spectrally negative Lévy process, the q-scale function starts at zero. The formal barrier candidate formula therefore vanishes at capital zero and barrier zero, even with an infinite derivative-zero-plus value. This closes the analytic half of the zero-barrier boundary package as a reusable named result, leaving only the stochastic dividend equality.

Preamble

import Theorems.Thm_AvramDividend_Classical_chat04_scale_zero_origin_of_unbounded_variation
import Theorems.Thm_AvramDividend_Classical_barrierValue_zero_of_scale_zero

open AvramDividend.Classical MeasureTheory
open scoped NNReal ENNReal

Formal statement

theorem AvramDividend.Classical.unbounded_variation_zero_barrier_candidate_vanishes
    {Ω : Type*} [mΩ : MeasurableSpace Ω]
    {P : Measure Ω} {𝓕 : Filtration ℝ≥0 mΩ}
    (X : SpectrallyNegativeLevy P 𝓕)
    (q : ℝ) (hq : 0 < q)
    (W : ℝ → ℝ) (hW : IsScaleFunction X q W)
    (hnbv : ¬ X.BoundedVariation) :
    barrierValue W 0 0 = 0 := by sorry

Source

Proved chat04_scale_zero_origin_of_unbounded_variation and barrierValue_zero_of_scale_zero.

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