Classic estimation methods for Hawkes processes rely on the assumption that observed event times are indeed a realisation of a Hawkes process, without considering any potential perturbation of the model. However, in practice, observations are often altered by some noise, the form of which depends on the context.It is then required to model the alteration mechanism in order to infer accurately such a noisy Hawkes process. While several models exist, we consider, in this work, the observations to be the indistinguishable union of event times coming from a Hawkes process and from an independent Poisson process. Since standard inference methods (such as maximum likelihood or Expectation-Maximisation) are either unworkable or numerically prohibitive in this context, we propose an estimation procedure based on the spectral analysis of second order properties of the noisy Hawkes process. Novel results include sufficient conditions for identifiability of the ensuing statistical model with exponential interaction functions for both univariate and bivariate processes. Although we mainly focus on the exponential scenario, other types of kernels are investigated and discussed. A new estimator based on maximising the spectral log-likelihood is then described, and its behaviour is numerically illustrated on synthetic data. Besides being free from knowing the source of each observed time (Hawkes or Poisson process), the proposed estimator is shown to perform accurately in estimating both processes.

翻译：霍克斯过程的经典估计方法依赖于观测事件时间确实是霍克斯过程实现这一假设，而未考虑模型的任何潜在扰动。然而在实际应用中，观测数据常受某种噪声干扰，其具体形式取决于应用场景。为准确推断此类含噪霍克斯过程，需建立噪声机制模型。尽管存在多种模型，本文考虑观测事件时间为霍克斯过程和独立泊松过程事件时间的不可区分并集。由于标准推断方法（如极大似然或期望最大化）在此背景下不可行或数值代价过高，我们提出基于含噪霍克斯过程二阶谱性质的估计流程。新结果包括：在指数型交互函数下，单变量及双变量过程统计模型可识别性的充分条件。虽重点探讨指数场景，亦研究并讨论其他类型核函数。随后描述基于最大化谱对数似然的新估计量，并在合成数据上数值验证其行为。该估计量无需区分每个观测时间的来源（霍克斯或泊松过程），在估计两个过程时均表现出精确性能。