%0 Journal Article %T Locally stationary Hawkes processes %+ Laboratoire Traitement et Communication de l'Information (LTCI) %+ Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA) %+ Mathématiques et Informatique Appliquées (MIA-Paris) %A Roueff, François %A von Sachs, Rainer %A Sansonnet, Laure %Z Projet d'Actions de Recherche Concertees" of the "Communaute francaise de Belgique" [12/17-045]; IAP research network Grant of the Belgian government (Belgian Science Policy) [P7/06]; Labex LMH [ANR-11-IDEX-003-02] %< avec comité de lecture %@ 0304-4149 %J Stochastic Processes and their Applications %I Elsevier %V 126 %N 6 %P Pages 1710–1743 %8 2016-06-01 %D 2016 %R 10.1016/j.spa.2015.12.003 %K point processes 2000 MSC: 60G55 %K Locally stationary processes %K Hawkes processes %K 46N30 %K 62M15 %K time frequency analysis %K Bartlett spectrum %Z Mathematics [math]/Statistics [math.ST]Journal articles %X 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. %G English %2 https://imt.hal.science/hal-01153882/document %2 https://imt.hal.science/hal-01153882/file/lshawkes.pdf %L hal-01153882 %U https://imt.hal.science/hal-01153882 %~ INSTITUT-TELECOM %~ AGROPARISTECH %~ INRA %~ INSMI %~ TELECOM-PARISTECH %~ PARISTECH %~ MIA-PARIS %~ AGREENIUM %~ AGROPARISTECH-SACLAY %~ INRA-SACLAY %~ AGROPARISTECH-MMIP %~ AGROPARISTECH-ORG %~ LTCI %~ IDS %~ S2A %~ INSTITUTS-TELECOM %~ INRAE %~ ANR %~ GS-COMPUTER-SCIENCE %~ TEST3-HALCNRS %~ MATHNUM