Bayesian Inference for Hawkes Processes With Stochastic Excitations

Classical Hawkes processes assume deterministic excitation decay, limiting their ability to capture the heterogeneity and randomness of influence magnitudes observed in real-world systems such as political violence, finance, and social interactions. While existing extensions have introduced randomness via independent and identically distributed excitation magnitudes, we focus on the case where excitation magnitudes are correlated over time, evolving stochastically via a Cox–Ingersoll–Ross diffusion process. This allows for time-varying, correlated excitation strengths, offering a more flexible model for complex event clustering. Due to the intractable likelihood introduced by the latent excitation mechanism, we explore and compare two Bayesian inference approaches: Markov Chain Monte Carlo (MCMC) and approximate Bayesian computation sequential Monte Carlo (ABC-SMC). Through comprehensive simulation studies, we find that ABC-SMC achieves more accurate parameter recovery than MCMC, particularly for the stochastic excitation parameters, with substantially lower root-mean-square error and mean absolute error. While MCMC exhibits systematic underestimation and struggles with identifiability, ABC-SMC provides more robust estimates, with credible intervals reliably covering true values. Our results highlight the practical advantage of simulation-based, likelihood-free methods for complex point process models with latent stochastic dynamics.