Efficient Simulation of Hawkes Processes using their Affine Volterra Structure

TL;DR

This paper presents a new, efficient simulation method for Hawkes processes that leverages their affine Volterra structure, enabling large-scale Monte Carlo simulations with deterministic complexity and high accuracy.

Contribution

The authors develop a novel simulation scheme for Hawkes processes based on their affine Volterra structure, offering deterministic complexity and broad kernel applicability.

Findings

Significant computational gains demonstrated in numerical experiments.

High accuracy maintained across various kernels.

Improved performance using the resolvent-based variant.

Abstract

We introduce a novel and efficient simulation scheme for Hawkes processes on a fixed time grid, leveraging their affine Volterra structure. The key idea is to first simulate the integrated intensity and the counting process using Inverse Gaussian and Poisson distributions, from which the jump times can then be easily recovered. Unlike conventional exact algorithms based on sampling jump times first, which have random computational complexity and can be prohibitive in the presence of high activity or singular kernels, our scheme has deterministic complexity which enables efficient large-scale Monte Carlo simulations and facilitates vectorization. Our method applies to any nonnegative, locally integrable kernel, including singular and non-monotone ones. By reformulating the scheme as a stochastic Volterra equation with a measure-valued kernel, we establish weak convergence to the target Hawkes process in the Skorokhod $J_1$-topology. Numerical experiments confirm substantial computational gains while preserving high accuracy across a wide range of kernels, with remarkably improved performance for a variant of our scheme based on the resolvent of the kernel.