The paper titled,”Spectra of some self-exciting and mutually exciting point processes”, is one of the most widely cited papers in marked point process literature. I guess this was the first paper that explored the complete covariance density function of point processes, and in particular, self exciting and mutually exciting processes. In the time series literature, the covariance of a stationary process at various lags have special meaning. If you consider the generating function of the covariance at various lags and evaluate at a specific complex exponential, you arrive at population spectrum of the time series.

The following note is motivated by the blog post, “Bitcoin Trade Arrival as Self-Exciting Process”. Since the author has shared the data and code, I wanted to check some of the numbers from the post. The author uses “ptproc” which is deprecated library and has been removed from CRAN. In this note I have used the trades dataset from the author’s github directory and fit a self-exciting model to the trade arrivals.

I have stumbled on to a few mini-projects that revolve around fitting univariate and bivariate Hawkes processes. In this post, I will briefly summarize the write ups : High Frequency Trade Prediction with Bivariate Hawkes Process The authors starts with a SDE for intensity process and formulate its solution as a univariate Hawkes process. A visual depiction of self-excited intensity process is obtained via simulation. The time change theorem is stated and a QQ plot of the compensator is shown to follow an exponential inter-arrival distribution.

The paper titled, “Self-Exciting Point Process Models of Civilian Deaths in Iraq”, deals with fitting point processes to civilian deaths from March 2003 to December 2007. In this post, I will summarize main points from the paper Firstly, What is “Operation Iraqi Freedom” ? Here’s a wiki blurb The 2003 invasion of Iraq lasted from 19 March to 1 May 2003 and signaled the start of the conflict that later came to be known as the Iraq War, which was dubbed Operation Iraqi Freedom by the United States.

The paper, written by Ioane Muni Toke and Fabrizio Pomponio, titled, Modeling Trades-Through in a Limit Order Book Using Hawkes Processes, uses Hawkes process to examine microstructure behavior. This paper uses Multivariate Hawkes process to model trades-through. The best thing about this paper is that the authors have made the dataset available for the readers so that they can work through the numbers and get a feel of model inference. The dataset is available at dataverse.