Multivariate Hawkes Processes (MHPs) are a class of point processes that can account for complex temporal dynamics among event sequences. In this work, we study the accuracy and computational efficiency of three classes of algorithms which, while widely used in the context of Bayesian inference, have rarely been applied in the context of MHPs: stochastic gradient expectation-maximization, stochastic gradient variational inference and stochastic gradient Langevin Monte Carlo. An important contribution of this paper is a novel approximation to the likelihood function that allows us to retain the computational advantages associated with conjugate settings while reducing approximation errors associated with the boundary effects. The comparisons are based on various simulated scenarios as well as an application to the study the risk dynamics in the Standard & Poor's 500 intraday index prices among its 11 sectors.

翻译：多元霍克斯过程（MHPs）是一类点过程，能够刻画事件序列间复杂的时序依赖关系。本研究系统评估了三类算法的精度与计算效率——随机梯度期望最大化、随机梯度变分推断及随机梯度朗之万蒙特卡洛——这些算法虽在贝叶斯推断领域广泛应用，但在MHPs场景下鲜有涉及。本文的重要贡献在于提出一种新颖的似然函数近似方法，该方法在保留共轭设置计算优势的同时，有效降低了边界效应带来的近似误差。比较分析基于多种模拟场景，并应用于标准普尔500指数日内价格中11个行业板块的风险动态研究。