Partial self-exciting point processes and their parameter estimations

Abstract

A partial self-exciting point process is proposed based on some phenomena in real world, which is a new kind of Hawkes process with partial self-exciting effect. The features, properties and moments of the partial self-exciting point process are investigated in terms of Dynkin formula and the method proposed by Cui et al. (2017 Cui, L., Hawkes, R. A. G. Yi. and H. 2017. A new method for moments of Hawkes processes. (submitted). [Google Scholar]), and the relationship between the new and conventional Hawkes processes is given. The maximum-likelihood estimations (MLEs) for the parameters in the partial self-exciting point process for two data situations are given via some sets of equations that can be solved numerically by suing the commend fslove in Matlab. Some numerical examples are presented based on simulation data to illustrate the MLE method, and an algorithm to generate the random event times for the partial self-exciting point process is presented as well. A primary goal of the work is to extend the applicability of Hawkes process and to present the MLE of parameters for partial self-exciting point process.