On the simulation of the Hawkes process via Lambert-W functions

TL;DR

This paper demonstrates that the inverse sampling transform for simulating Hawkes processes can be effectively expressed using Lambert-W functions, leading to a more efficient implementation than recent methods.

Contribution

It introduces a Lambert-W function-based formulation of the inverse sampling transform, improving computational efficiency in Hawkes process simulation.

Findings

Lambert-W functions facilitate the inverse sampling transform.

The proposed method outperforms recent simulation techniques.

Efficient implementation reduces computational time.

Abstract

Several methods have been developed for the simulation of the Hawkes process. The oldest approach is the inverse sampling transform (ITS) suggested in \citep{ozaki1979maximum}, but rapidly abandoned in favor of more efficient alternatives. This manuscript shows that the ITS approach can be conveniently discussed in terms of Lambert-W functions. An optimized and efficient implementation suggests that this approach is computationally more performing than more recent alternatives available for the simulation of the Hawkes process.