The Poisson process

Summary

This chapter serves as an interlude between the discrete-time theory we just reviewed and the continuous-time theory. The Poisson process being probably the best (for pedagogical purposes) prototype of continuous-time Markov process, it provides a good occasion to gently introduce the concepts specific to this framework (such as right constancy, jump times, jump chain, properties of increments) before hitting the reader with the general theory, which will be done in the next chapter. At the heart of our approach to time-continuous processes is time discretization; the proofs of Theorems 3.9 and 3.10 will provide a gentle introduction to this method, which will be used again in the next chapter (with unfortunately many technical complications in the general case, where increments are not assumed to be stationary).