Reverse-time scores and convex-order monotonicity for additive processes with Lévy clocks

Valentin Dendoncker, Michel Denuit () and Christian Y. Robert
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Valentin Dendoncker: Université catholique de Louvain, LIDAM/ISBA, Belgium
Michel Denuit: Université catholique de Louvain, LIDAM/ISBA, Belgium

No 2026023, LIDAM Discussion Papers ISBA from Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA)

Abstract: We study reverse-time conditional means, scores and drift quantities in additive noisemodels driven by deterministic Lévy clocks. Starting from an integrable signal Y and an independent centered additive process (Zt)t∈[0,T ], we consider the forward noising model Xt = Y + Zt, 0 ≤t≤T, and its reverse-time version Xs = XT−s. The main examples are Gaussian noise with deterministic variance clock, compensated compound Poisson noise with deterministic intensity clock, and symmetric α-stable noise with stable clock, for α∈(1,2). We first derive Bartlett–Tweedie identities for additive noise models and characterize, within the infinitely divisible class, the cases in which the denoising map x→E[Y |Y + ε= x] is affine. This occurs precisely when the signal and the noise are compatible with the same convolution semigroup. We then prove that the reverse conditional mean and the denoising error normalized by the accumulated noise clock are martingales in reverse time. As a consequence, both families are increasing in the convex order along reverse time. The actual reverse drift, or reverse local mean drift in the jump and stable cases, is obtained by multiplying this normalized score by the instantaneous clock rate. It is therefore not a martingale in general, but it defines a peacock when the instantaneous rate is non-decreasing in reverse time. Explicit linear formulas are obtained for semigroup-compatible initial laws. The Cauchy case is treated separately, since the lack of integrability prevents an ordinary convex-order analysis of the reverse score drift.

Keywords: Conditional expectation; convex order; peacocks; reverse-time dynamics; additive processes; Lévy clocks; infinitely divisible distributions; Tweedie formula (search for similar items in EconPapers)
Pages: 40
Date: 2026-06-08
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Persistent link: https://EconPapers.repec.org/RePEc:aiz:louvad:2026023

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