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

2025-07-21

AI Summary

This paper addresses spatiotemporal epidemic dynamics modeling via a generalized multivariate Hawkes process featuring time-varying baseline intensities and nonparametric triggering kernels—inducing inherent non-Markovianity. Method: We derive, for the first time, a closed-form expression for the joint characteristic function at multiple time points. Leveraging functional Laplace transforms and integral equation theory, we establish a system of distributional equations for non-Markovian Hawkes processes and analytically decompose the temporal evolution of the first two moments and cross-time covariances—including both singular and absolutely continuous components. Combining infinite divisibility with numerical approximation techniques, we enable statistical inference and simulation for high-dimensional processes. Results: Empirical validation confirms the theoretical correctness of our formulations and demonstrates high simulation fidelity, substantially extending the applicability of Hawkes processes to complex, realistic epidemic propagation modeling.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Extend multivariate Hawkes processes for spatio-temporal epidemic modeling
• Derive closed-form expressions for characteristic functions and moments
• Develop numerical approximations for Laplace transform and moments

Innovation

• Closed-form multivariate multi-temporal characteristic function derivation
• Infinitely divisible property for probability distribution equations
• Analytical decomposition of covariance into singular and continuous parts