  Locally stationary Hawkes processes have been introduced in order to
generalise classical Hawkes processes away from stationarity by allowing for a
time-varying second-order structure. This class of self-exciting point
processes has recently attracted a lot of interest in applications in the life
sciences (seismology, genomics, neuro-science,...), but also in the modelling
of high-frequency financial data. In this contribution we provide a fully
developed nonparametric estimation theory of both local mean density and local
Bartlett spectra of a locally stationary Hawkes process. In particular we apply
our kernel estimation of the spectrum localised both in time and frequency to
two data sets of transaction times revealing pertinent features in the data
that had not been made visible by classical non-localised approaches based on
models with constant fertility functions over time.
