Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation

Isaias Banales; Arturo Jaramillo

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arXiv · 2610.19588

Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation

Isaias Banales·Arturo Jaramillo

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Abstract

We consider the problem of estimating the background intensity of a Hawkes process when the triggering mechanism is known, treating the background as an unknown finite positive measure with a regular density. We propose a likelihood-based variational approach based on the Wasserstein-Fisher-Rao geometry. In this setting, the relevant first variations are naturally distribution-valued and therefore fall outside the classical gradient-flow formulation. We develop a distributional framework for the resulting dynamics and give them a finite-dimensional meaning through projections onto statistical manifolds. We identify the true background as the unique zero of a limiting generalized gradient and, under suitable stability, compactness, integrability and approximation conditions, establish a conditional consistency theorem for measurable estimators satisfying the empirical projected equations on events whose probabilities tend to one. Finally, we specialize the construction to Gaussian-mixture manifolds and obtain an explicit finite-dimensional scheme.

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Isaias Banales, Arturo Jaramillo. 2026-10-14. Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation. https://arxiv.org/abs/2610.19588

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