I have to do some work where I need to estimate the parameters for a poisson process and a Hawkes process from data. I was looking through some of my old probability and stochastic processes textbooks, but I really could not find much on the actual parameter estimation of these process. I checked Grimmett and Stirzaker and the Sheldon Ross' Probability Models book, but they don't reference much on parameter estimation, etc. The Vere-Jones book on point processes as a bit more info, but not really fully fleshed examples.
It seems a bit odd that there is not more on numerical methods and estimation of parameters for stochastic processes? Since there is so much theory on the numerical issues surrounding numerical estimation of say ordinary differential equations and partial differential equations, I assume numerical estimation would be a popular topic for stochastic processes too. There are a number of books on estimating Stochastic Differential Equations, but they don't seem to cover other stochastic processes--as far as I could see. I am curious if I am just looking in the wrong places, or if there is just a lack of materials?
Seems like maximum likelihood would be a simple enough method to fit parameters for some of these models, but there could be numerical computing issues that I am not thinking of. I can certainly envision numerical computing issues cropping up in the case of continuous time stochastic processes--that need to be approximated by some discrete scheme. What about having lots of zeros, or the potential for big jumps--these things might generally throw off the optimization routine for a given model.
So any good reference on parameter estimation for different types of stochastic processes would really be appreciated. Thanks.