Ergodicity and scaling limit of a constrained multivariate Hawkes process

TL;DR

This paper introduces a constrained multivariate Hawkes process with boundary conditions, analyzes its ergodic properties, and derives scaling limits, with applications to high-frequency financial data modeling.

Contribution

It presents a novel constrained Hawkes process model, establishes conditions for ergodicity, and derives its scaling limits, extending the understanding of such processes in financial applications.

Findings

Conditions for ergodicity of the process are established.

Scaling limits for the integrated point process are derived.

The model is applicable to high-frequency financial data analysis.

Abstract

We introduce a multivariate Hawkes process with constraints on its conditional density. It is a multivariate point process with conditional intensity similar to that of a multivariate Hawkes process but certain events are forbidden with respect to boundary conditions on a multidimensional constraint variable, whose evolution is driven by the point process. We study this process in the special case where the fertility function is exponential so that the process is entirely described by an underlying Markov chain, which includes the constraint variable. Some conditions on the parameters are established to ensure the ergodicity of the chain. Moreover, scaling limits are derived for the integrated point process. This study is primarily motivated by the stochastic modelling of a limit order book for high frequency financial data analysis.