At the opposite extreme from Brownian motion is the Poisson process. This is a process that only changes value by means of jumps, and even then, the jumps are nicely spaced. The Poisson process is the prototype of a pure jump process, and later we will see that it is the building block for an important class of stochastic processes known as Lévy processes.

Definition 5.1 Let {ℱt} be a filtration, not necessarily satisfying the usual conditions. A Poisson process with parameter λ > 0 is a stochastic process X satisfying the following properties:

1. (1) X0 = 0, a.s.

2. (2) The paths of Xt are right continuous with left limits.

3. (3) If s < t, then Xt – Xs is a Poisson random variable with parameter λ(t − s).

4. (4) If s < t, then Xt – Xs is independent of ℱs.

Define Xt− = lims→t,s<tXs, the left-hand limit at time t, and ΔXt = Xt – Xt–, the size of the jump at time t. We say a function f is increasing if s < t implies f(s) ≤ f(t). We use “strictly increasing” when s < t implies f(s) < f(t). We have the following proposition.

Proposition 5.2Let X be a Poisson process. With probability one, the paths of Xt are increasing and are constant except for jumps of size 1. There are only finitely many jumps in each finite time interval.