Poisson Processes

The standard Poisson process is introduced and analyzed in all its peculiar aspects, characteristics and properties. Different definitions are given and proved to be equivalent. The concepts of overdispersion and contagion are introduced into the Poisson model. Other analyzed generalizations include the weighted Poisson process, the Skellam process, the time non-homogeneous Poisson process, with the intensity parameter time dependent, the compound Poisson process, and some more advanced models including the doubly stochastic Poisson process, the iterated Poisson and the shot-noise and Hawkes processes. Along the chapter, the estimation methods (moments and maximum likelihood) are addressed to show how to implement the models with real or simulated data. Many examples, from horse kicks to wars, car accidents, fire insurance and game scores, are provided.