5 - The Poisson process

Summary

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 {ℱ_t} 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 X_t are right continuous with left limits.

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

4. (4) If s < t, then X_t – X_s is independent of ℱ_s.

Define X_t− = lim_s→t,s<tX_s, the left-hand limit at time t, and ΔX_t = X_t – X_t–, 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 X_t are increasing and are constant except for jumps of size 1. There are only finitely many jumps in each finite time interval.