An algorithm for the computation of joint Hawkes moments with exponential kernel

TL;DR

This paper introduces a recursive algorithm implemented in Maple and Mathematica for efficiently computing joint moments and cumulants of Hawkes processes with exponential kernels, addressing computational complexity issues.

Contribution

It presents a novel recursive algorithm for calculating joint moments and cumulants of Hawkes processes with exponential kernels, with implementations in Maple and Mathematica.

Findings

Algorithm effectively computes high-order moments and cumulants.

Numerical results demonstrate manageable computation times.

Closed-form expressions are complex, with thousands of summands for higher orders.

Abstract

The purpose of this paper is to present a recursive algorithm and its implementation in Maple and Mathematica for the computation of joint moments and cumulants of Hawkes processes with exponential kernels. Numerical results and computation times are also discussed. Obtaining closed form expressions can be computationally intensive, as joint fifth cumulant and moment formulas can be respectively expanded into up to 3,288 and 27,116 summands.