Time-frequency analysis of locally stationary Hawkes processes

TL;DR

This paper develops a nonparametric estimation framework for locally stationary Hawkes processes, enabling detailed time-frequency analysis of complex data, with applications in finance and sciences.

Contribution

It introduces a novel kernel-based method for estimating local spectra and mean density of locally stationary Hawkes processes, advancing nonparametric analysis techniques.

Findings

Effective local spectrum estimation reveals hidden data features.

Application to financial transaction data uncovers new insights.

Method outperforms classical approaches with constant parameters.

Abstract

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.