Convergence of the Discrete-Time Compound Hawkes Processwith Exponential or Erlang Kernel

TL;DR

This paper introduces a discrete-time Hawkes process model with exponential or Erlang kernels, proves its convergence to the continuous-time Hawkes process as the time step decreases, and highlights its applicability to regularly sampled data.

Contribution

It presents a novel discrete-time Hawkes process model and establishes its convergence to the continuous-time counterpart, bridging a gap for regularly sampled data analysis.

Findings

The discrete-time model converges to the continuous-time Hawkes process as the time step approaches zero.

The model is applicable to data sampled at regular intervals.

Theoretical proof of convergence enhances understanding of discrete and continuous Hawkes processes.

Abstract

Due to its clustering and self-exciting properties, the Hawkes process has been used extensively in numerous fields ranging from sismology to finance. Since data is often aquired on regular time intervals, we propose a piece-wise constant model based on a Discrete-Time Hawkes Process (DTHP). We prove that this discrete-time model converges to the usual continuous-time Hawkes process as the time-step tends to zero.