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（Multivariate Hawkes Processes, MHPs）是一类能够刻画事件序列间复杂时序动态的点过程。本研究重点分析三类算法的精度与计算效率——随机梯度期望最大化、随机梯度变分推断以及随机梯度朗之万蒙特卡洛——这些方法在贝叶斯推断中广泛应用，但在MHP领域鲜有探究。本文的重要贡献在于提出一种似然函数的新型近似方法，该方法既能保持共轭设置下的计算优势，又可降低边界效应导致的近似误差。我们基于多种模拟场景及标准普尔500指数11个行业日内价格风险动态的实际应用开展对比研究。