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 times of a diffusion in a continuous path space to new arrivals of the point process. This leads to a continuous sample path that is used to describe the underlying mechanism generating the arrival distribution. 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 propose methods for inferring and sampling from the point process derived from the latent diffusion process. We illustrate the approach with numerical examples using both simulated and real data. The proposed methods and framework provide a basis for interpreting point processes through the lens of diffusions.

翻译：点过程通常与连续过程存在自然的解释关系。我们提出一种点过程构造方法，将到达时间观测值描述为潜在扩散过程状态的表现。在该框架下，我们将连续路径空间中扩散的返回时间与点过程的新到达事件相关联，由此生成用于描述到达分布产生机制的连续样本路径。此类模型出现在多个学科领域，例如金融市场中由隐藏连续价格决定的市场行为，或神经科学中潜在刺激产生脉冲序列的情形。基于伊藤游程理论的最新进展，我们提出了从潜在扩散过程推导出的点过程推断与采样方法。通过数值实例（包括模拟数据与真实数据）展示了该方法的有效性。本文提出的方法与框架为通过扩散过程视角解释点过程提供了理论基础。