Locally stationary Hawkes processes

François Roueff (1) , Rainer von Sachs (2) , Laure Sansonnet (3)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA)
3 MIA-Paris - Mathématiques et Informatique Appliquées

This paper addresses the generalisation of stationary Hawkes processes in order to allow for a time-evolving second-order analysis. Motivated by the concept of locally stationary autoregressive processes, we apply however inherently different techniques to describe the time-varying dynamics of self-exciting point processes. In particular we derive a stationary approximation of the Laplace transform of a locally stationary Hawkes process. This allows us to define a local intensity function and a local Bartlett spectrum which can be used to compute approximations of first and second order moments of the process. We complete the paper by some insightful simulation studies.

Keywords: point processes 2000 MSC: 60G55; Locally stationary processes; Hawkes processes; 46N30; 62M15; time frequency analysis; Bartlett spectrum

Domains: Mathematics [math]; Statistics [math.ST]

François Roueff, Rainer von Sachs, Laure Sansonnet. Locally stationary Hawkes processes. Stochastic Processes and their Applications, 2016, 126 (6), pp.Pages 1710-1743. ⟨10.1016/j.spa.2015.12.003⟩. ⟨hal-01153882⟩