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Efficient Simulation of Hawkes Processes using their Affine Volterra Structure

Eduardo Abi Jaber, Elie Attal, Dimitri Sotnikov

TL;DR

The paper addresses efficient simulation of Hawkes processes with non-Markovian kernels by introducing the Hawkes iVi scheme, which operates on a fixed time grid to simulate the integrated intensity $\Lambda$ and the counting process $N$ using Inverse Gaussian increments and Poisson counts. The method replaces the random, quadratic complexity of traditional jump-time-first schemes with a deterministic $\mathcal{O}(n^2)$ approach, enabling vectorization and scalable Monte Carlo simulations while accommodating general kernels including singular and non-monotone ones. A key theoretical contribution is the weak convergence of the discretized scheme $(\Lambda^n,N^n)$ to the true Hawkes process $(\Lambda,N)$ in the Skorokhod $J_1$ topology, proven by recasting the scheme as a stochastic Volterra equation with a measure-valued kernel and establishing tightness and limit identification. The paper also presents practical enhancements, including a resolvent-augmented version that accelerates convergence and a Markovian variant for exponential kernels, with extensive numerical experiments demonstrating substantial computational gains and high accuracy across kernel choices. Overall, the Hawkes iVi framework provides a robust, deterministic, and scalable tool for large-scale Monte Carlo simulations of Hawkes processes in diverse applications.

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.

Efficient Simulation of Hawkes Processes using their Affine Volterra Structure

TL;DR

The paper addresses efficient simulation of Hawkes processes with non-Markovian kernels by introducing the Hawkes iVi scheme, which operates on a fixed time grid to simulate the integrated intensity and the counting process using Inverse Gaussian increments and Poisson counts. The method replaces the random, quadratic complexity of traditional jump-time-first schemes with a deterministic approach, enabling vectorization and scalable Monte Carlo simulations while accommodating general kernels including singular and non-monotone ones. A key theoretical contribution is the weak convergence of the discretized scheme to the true Hawkes process in the Skorokhod topology, proven by recasting the scheme as a stochastic Volterra equation with a measure-valued kernel and establishing tightness and limit identification. The paper also presents practical enhancements, including a resolvent-augmented version that accelerates convergence and a Markovian variant for exponential kernels, with extensive numerical experiments demonstrating substantial computational gains and high accuracy across kernel choices. Overall, the Hawkes iVi framework provides a robust, deterministic, and scalable tool for large-scale Monte Carlo simulations of Hawkes processes in diverse applications.

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 -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.

Paper Structure

This paper contains 26 sections, 13 theorems, 154 equations, 11 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Connection to the literature.
  3. Outline of the paper.
  4. Notation.
  5. Deriving the Hawkes iVi scheme
  6. Approximating the dynamics
  7. Deriving the Inverse Gaussian parameters
  8. Weak convergence of the Hawkes iVi scheme
  9. Numerical results
  10. Population approach and Ogata's algorithm
  11. Population approach
  12. Ogata's algorithm
  13. Numerical experiments
  14. Laplace transforms.
  15. Comparison of marginal laws.
  16. ...and 11 more sections

Key Result

Theorem 3.1

We have the weak convergence in the Skorokhod $J_1$-topology, where $N$ is a Hawkes process on $[0\,,T]\,$ with exogenous intensity $g_0$ and memory kernel $K$, and $\Lambda$ is its integrated intensity.

Figures (11)

  • Figure 1: Sample trajectories simulated with the iVi scheme with time step $0.001$.
  • Figure 2: Convergence of the Laplace transform with $w = -\frac{1}{\mathbb{E}[N_T]}$ for $N_T$ (left) and $\Lambda_T$ (right). The $x$-axis shows the number of time steps, and the $y$-axis shows the absolute error of the Monte Carlo estimator.
  • Figure 3: Simulation time for $10^5$ trajectories ($y$-axis) versus the absolute error of the Monte Carlo estimator of the Laplace transform ($x$-axis). Crosses represent estimators obtained with the iVi scheme under different discretization steps. Horizontal bars indicate the exact methods: Population (blue) and Ogata (purple).
  • Figure 4: Marginal law of $N_T$. Empirical densities (top left), distances to the empirical CDF of the population method (top right), Q-Q plots for Ogata algorithm (bottom left) and iVi scheme (bottom right). A time step $0.01$ was taken for the iVi scheme.
  • Figure 5: Marginal law of $\Lambda_T$. Empirical densities (top left), distances to the emperical CDF of the population method (top right), Q-Q plots for Ogata algorithm (bottom left) and iVi scheme (bottom right). A time step $0.01$ was taken for the iVi scheme.
  • ...and 6 more figures

Theorems & Definitions (27)

  • Theorem 3.1
  • proof
  • Theorem 4.1: Random time change
  • Proposition 4.2
  • proof
  • Remark 4.3
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 17 more