Quasi likelihood analysis of point processes for ultra high frequency data

TL;DR

This paper develops a quasi likelihood framework for point process models, including Hawkes processes, applicable to high-frequency financial data, and analyzes their large-sample properties under non-ergodic conditions.

Contribution

It introduces a generalized stochastic regression model for point processes and establishes large deviation and asymptotic properties of estimators in high-intensity regimes.

Findings

Established polynomial large deviation estimates.

Derived asymptotic properties of estimators in high-intensity settings.

Addressed non-ergodic statistical behavior.

Abstract

We introduce a point process regression model that is applicable to price models and limit order book models. Hawkes type autoregression in the intensity process is generalized to a stochastic regression to covariate processes. We establish the so-called quasi likelihood analysis, which gives a polynomial type large deviation estimate for the statistical random field. We derive large sample properties of the maximum likelihood type estimator and the Bayesian type estimator when the intensity processes become large under a finite time horizon. There appears non-ergodic statistics. A classical approach is also mentioned.