A counting process, X, is defined to be an adapted stochastic process starting from zero which is piecewise constant and right-continuous with jumps of size 1. That is, letting be the first time at which , then
By the debut theorem, are stopping times. So, X is an increasing integer valued process counting the arrivals of the stopping times . A basic example of a counting process is the Poisson process, for which has a Poisson distribution independently of , for all times , and for which the gaps between the stopping times are independent exponentially distributed random variables. As we will see, although Poisson processes are just one specific example, every quasi-left-continuous counting process can actually be reduced to the case of a Poisson process by a time change. As always, we work with respect to a complete filtered probability space .
Note that, as a counting process X has jumps bounded by 1, it is locally integrable and, hence, the compensator A of X exists. This is the unique right-continuous predictable and increasing process with such that is a local martingale. For example, if X is a Poisson process of rate , then the compensated Poisson process is a martingale. So, the compensator of X is the continuous process . More generally, X is said to be quasi-left-continuous if for all predictable stopping times , which is equivalent to the compensator of X being almost surely continuous. Another simple example of a counting process is for a stopping time , in which case the compensator of X is just the same thing as the compensator of .
As I will show in this post, compensators of quasi-left-continuous counting processes have many parallels with the quadratic variation of continuous local martingales. For example, Lévy’s characterization states that a local martingale X starting from zero is standard Brownian motion if and only if its quadratic variation is . Similarly, as we show below, a counting process is a homogeneous Poisson process of rate if and only if its compensator is . It was also shown previously in these notes that a continuous local martingale X has a finite limit if and only if is finite. Similarly, a counting process X has finite value at infinity if and only if the same is true of its compensator. Another property of a continuous local martingale X is that it is constant over all intervals on which its quadratic variation is constant. Similarly, a counting process X is constant over any interval on which its compensator is constant. Finally, it is known that every continuous local martingale is simply a continuous time change of standard Brownian motion. In the main result of this post (Theorem 5), we show that a similar statement holds for counting processes. That is, every quasi-left-continuous counting process is a continuous time change of a Poisson process of rate 1. Continue reading “Compensators of Counting Processes”