多元Hawkes过程的泛函拉普拉斯变换、后续特征与数值近似
Functional Laplace Transform of a Multivariate Hawkes Process, Subsequent Characteristics, and Numerical Approximations
AI总结：
本文针对一般多元Hawkes过程，给出多元多时刻特征函数、计数概率分布方程及前两阶矩和协方差函数的闭式表达，并提出拉普拉斯变换与矩的数值近似方案。
摘要：
基于Hawkes过程的大量研究已在金融、生物学和社交网络等许多领域展开。Hawkes过程构成一类自激发的简单点过程。本文考虑一类一般的多元Hawkes过程，旨在对时空流行病的动态进行建模。对于该类过程，触发基线强度是时间依赖的，且激发矩阵函数是一般形式的，这使得模型在大多数情况下是非马尔可夫的。本文首先给出这些Hawkes过程的多元多时刻特征函数的闭式表达式，以自然的方式推广了Hawkes文献中经典的单时刻公式。然后，利用Hawkes过程的无穷可分性质，推导出与每个单时刻计数概率分布相关的方程组，并使其适用于本文所考虑的一般形式的Hawkes模型。接下来，给出过程前两阶矩的时间结构的闭式公式，由此推导出两个不同时刻多元协方差函数的一个原始表达式，从而扩展了针对更受限Hawkes过程类已有的结果。基于该表达式，我们将两个不同时刻的协方差解析分解为奇异部分和连续部分。最后给出简要的数值内容：提出一个用于拉普拉斯变换和前两阶矩数值近似的简单方案，并给出不同相关积分方程解的示例。还提供了不同模型设定下多元Hawkes过程的示例性模拟。
英文摘要：
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.
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