T.Ozaki’s paper titled, “Maximum likelihood estimation of Hawkes' self-exciting point processes”, deals with the univariate Hawkes’ process. The paper gives a detailed method to obtain the ML estimates of the process. In order to verify the ML estimates, the author simulates the process and compares the true vs. estimated parameters. I found a typo in one of the expressions for the gradient. In any case, R provides nlm function, that computes the gradient and hessian numerically. So, if one chooses to, one can slog out the expressions for gradient and hessian, and then feed in to nlm , else, you can allow the function to do the job numerically.

The following document contains some R code that I have written to estimate the parameters for Hawkes’ process. A good way to check whether the parameter estimation has gone right is by plotting a particular time changed process that should result in standard Poisson. It is easy to work out the compensator, given that the paper gives a clear explanation of all the necessary expressions.

MLE of Hawkes Self-Exciting Process