Keywords: point process, brownian excursion, diffusion
TL;DR: We develop methods to represent point processes in terms of excursions of a diffusion.
Abstract: Point processes often have a natural interpretation with respect to a continuous process. We propose a point process construction that describes arrival time observations in terms of the state of a latent diffusion process. In this framework, we relate the return time of diffusion in a continuous path space to new arrivals of the point process. These models arise in many disciplines, such as financial settings where actions in a market are determined by a hidden continuous price or in neuroscience where a latent stimulus generates spike trains. Based on the developments in It\^o's excursion theory, we describe computational methods for inferring and sampling from the point process derived from the diffusion process. We provide numerical examples for the proposed method using both simulated and real data to illustrate the approach. The proposed methods and framework provide a basis for interpreting point processes through the lens of a diffusion.
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