Nonstationary Marked Hawkes processes in the high intensity regime

摘要:We study marked Hawkes processes with an intensity process which has  a non-stationary baseline intensity, and a general self-exciting  function of event ages at each time and marks. The marks are assumed to  be conditionally independent given the even times, while the  distribution of each mark depends on the event time, that is,  time-varying. We first observe an immigration-birth (branching)  representation of such a nonstationary marked Hawkes process, and then  derive equivalent representation of the process using the associated  conditional inhomogeneous Poisson processes with stochastic intensities.  We consider such a Hawkes process in the high intensity regime, where  the baseline intensity is scaled up, while the self-exciting function  and distributions of the marks are unscaled, and there is no  time-scaling in the scaled Hawkes process. We prove functional law of  large numbers and functional central limit theorems (FCLT) for the  scaled Hawkes processes in this asymptotic regime. The limits in the  FCLT are characterized by continuous Gaussian processes with covariance  structures expressed with convolution functionals resulting from the  branching representation. 

报告人简介:李波2009年毕业于南开大学概率论与数理统计专业,现任南开大学数学科学学院数理金融与精算科学系副教授。主要的研究领域是随机过程理论及其在保险精算中的应用。目前研究兴趣是 与谱负Levy过程相关的位势理论、以及过程极限。  


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