Recursive computation of the Hawkes cumulants

TL;DR

This paper introduces a recursive, combinatorial method for efficiently computing higher-order cumulants of Hawkes processes, simplifying implementation compared to traditional differential equation or martingale approaches.

Contribution

It presents a novel recursive approach using Bell polynomials for calculating cumulants of Hawkes processes, applicable to multidimensional cases.

Findings

Method is easier to implement for higher-order cumulants.

Results are validated through Monte Carlo simulations.

Applicable to joint cumulants in multidimensional processes.

Abstract

We propose a recursive method for the computation of the cumulants of self-exciting point processes of Hawkes type, based on standard combinatorial tools such as Bell polynomials. This closed-form approach is easier to implement on higher-order cumulants in comparison with existing methods based on differential equations, tree enumeration or martingale arguments. The results are corroborated by Monte Carlo simulations, and also apply to the computation of joint cumulants generated by multidimensional self-exciting processes.