标题： 多元霍克斯过程的函数拉普拉斯变换、后续特征及数值近似 显示英文标题

标题： Functional Laplace Transform of a Multivariate Hawkes Process, Subsequent Characteristics, and Numerical Approximations

摘要： 大量基于霍克斯过程的研究已在金融、生物和社会网络等多个领域展开。 霍克斯过程形成了一类自激发的简单点过程。 在本文中，我们考虑了一类广义的多变量霍克斯过程，旨在模拟时空流行病的动力学。 对于这类过程，点燃的基础强度是时间相关的，而激发矩阵函数是通用的，使得模型在大多数情况下是非马尔可夫的。 在本文中，我们首先提供了这些霍克斯过程的多变量多时间特征函数的显式表达式，以自然的方式扩展了霍克斯文献中发现的经典单时间公式。 然后，我们利用霍克斯过程的无限可分性，推导出每个单时间计数概率分布的相关方程组，适用于本文中考虑的霍克斯模型的一般形式。 接下来，我们提供了该过程前两个时刻的时序结构的显式公式，这使我们能够推导出两个不同时间点的多变量协方差函数的原始表达式，从而扩展了现有针对更受限的霍克斯过程类别的结果。 基于此表达式，我们对两个不同时间点的协方差进行了分析分解，分为奇异部分和连续部分。 最后，我们给出了一些简要的数值元素：我们提出了一种用于拉普拉斯变换和前两个矩的数值近似的简单方案，并给出了不同相关积分方程的解的例子。 我们还提供了不同模型规范下的多变量霍克斯过程的示例模拟。 显示英文摘要

摘要： Numerous studies grounded on Hawkes processes have been carried out in many fields including finance, biology and social network. Hawkes processes form a class of selfexciting simple point processes. In this article, we consider a general class of multivariate Hawkes processes envisioned to model dynamics of spatio-temporal epidemics. For this class, the igniting baseline intensity is time dependent and the exciting matrix function is a general one, making the model non-Markovian in most of the cases. In this article, we first provide the closed-form expression of the multivariate multi-temporal characteristic function of these Hawkes processes, extending in a natural way the classical single-time formula found in the Hawkes literature. Then, we use the infinitely divisible property of the Hawkes process to derive the equation system related to the probability distribution of counts at each single time, adapted to the general formulation of the Hawkes model considered in this article. Next, we provide closed-form formulas for the temporal structure of the two first moments of the process, which allows us to deduce an original expression of the multivariate covariance function at two distinct times, thereby extending existing results established for more restricted classes of Hawkes processes. Based on this expression, we analytically decompose the covariance at two distinct times into singular and continuous parts. We finish with brief numerical elements: We present a simple scheme for numerical approximations of the Laplace transform and the first two moments, and give examples of solutions of the different related integral equations. We also provides illustrative simulations of the multivariate Hawkes process for different model specifications.