The building blocks of stochastic calculus — Part II: Poisson processes
In my previous article https://oscarnieves100.medium.com/the-building-blocks-of-stochastic-calculus-part-i-d06c87916070 I discussed the first building block of stochastic calculus: the Wiener process W(t). The Wiener process is useful for modelling a whole bunch of continuous random processes which are normally distributed, or which are made up of normal variables.
By contrast, when we want to model discrete random processes, the Poisson process J(t) is very useful and one which is actively used for modelling so-called “jump processes” (e.g. random processes with sudden jumps in them).
Functions of J(t)
Let us begin our discussion by recalling the probability mass function of J(t)
which has mean λ. In a general sense, J(t) represents the number of arrivals or successes x in a finite interval of duration t. The “jumpyness” of J(t) can be more easily observed in Figure 1 below, for a process X(t) which is the cumulative sum of finite steps dJ(t).
The MGF of J(t) can be found as follows:
