Gradient-based estimation of linear Hawkes processes with general kernels

Abstract

Linear multivariate Hawkes processes (MHP) are a fundamental class of point processes with self-excitation. When estimating parameters for these processes, a difficulty is that the two main error functionals, the log-likelihood and the least squares error (LSE), as well as the evaluation of their gradients, have a quadratic complexity in the number of observed events. In practice, this prohibits the use of exact gradient-based algorithms for parameter estimation. We construct an adaptive stratified sampling estimator of the gradient of the LSE. This results in a fast parametric estimation method for MHP with general kernels, applicable to large datasets, which compares favourably with existing methods.

Keywords: Hawkes processes, stochastic gradient descent, point processes, Monte Carlo methods, adaptive stratified sampling.

JEL Classification: C13, C14, C22