论文标题
霍克斯过程的理论分析和模拟方法及其扩散近似
Theoretical analysis and simulation methods for Hawkes processes and their diffusion approximation
论文作者
论文摘要
Ditlevsen（2017）引入了与Erlang记忆内核相互作用的鹰派过程的振荡系统。它们是分段确定性的马尔可夫过程（PDMP），可以通过随机扩散来近似。首先，证明了PDMP和扩散之间的强误差。其次，得出所得扩散的矩边界。第三，提出了基于数值分裂方法的扩散方案。事实证明，这些方案与均方顺序1融合，并保留扩散的特性，特别是低纤维化，奇迹性和力矩边界。最后，通过数值实验比较PDMP和扩散，其中PDMP通过适应的稀疏过程模拟。
Oscillatory systems of interacting Hawkes processes with Erlang memory kernels were introduced in Ditlevsen (2017). They are piecewise deterministic Markov processes (PDMP) and can be approximated by a stochastic diffusion. First, a strong error bound between the PDMP and the diffusion is proved. Second, moment bounds for the resulting diffusion are derived. Third, approximation schemes for the diffusion, based on the numerical splitting approach, are proposed. These schemes are proved to converge with mean-square order 1 and to preserve the properties of the diffusion, in particular the hypoellipticity, the ergodicity and the moment bounds. Finally, the PDMP and the diffusion are compared through numerical experiments, where the PDMP is simulated with an adapted thinning procedure.
