In mathematical finance the most important stochastic process is the Wiener process, which is used to model continuous asset price paths. The next important stochastic process is the Poisson process, used to model discontinuous random variables. Although time is continuous, the variable is discontinuous where it can represent a “jump” in an asset price (e.g., electricity prices or a credit risk event, such as describing default and rating migration scenarios). In this chapter we will discuss the Poisson process and some generalisations of it, such as the compound Poisson process and the Cox process (or doubly stochastic Poisson process) that are widely used in credit risk theory as well as in modelling energy prices.
In this section, before we provide the definition of a Poisson process, we first define what a counting process is.