The rough Hawkes process

This article studies the properties of Hawkes process with a gamma memory kernel and a shape parameter 𝛼∈(0,1]. This process, called rough Hawkes process, is nearly unstable since its intensity diverges to +∞ for a very brief duration when a jump occurs. First, we find conditions that ensure the stability of the process and provide a closed form expression of the expected intensity. Second, we next reformulate the intensity as an infinite dimensional Markov process. Approximating these processes by discretization and then considering the limit leads to the Laplace transform of the point process. This transform depends on the solution of an elegant fractional integro-differential equation. The fractional operator is defined by the gamma kernel and is similar to the left-fractional Riemann-Liouville integral. We provide a simple method for computing the Laplace transform. This is easily invertible by discrete Fourier transform so that the probability density of the process can be recovered. We also propose two methods of simulation. We conclude the article by presenting the log-likelihood of the rough Hawkes process and use it to fit hourly Bitcoin log-returns from 9/2/18 to 9/2/23.