Poisson Processes and Their Mixtures

Let us now move in the opposite direction from the continuous sample path Brownian motion model considered in the previous chapter, and consider a stochastic process {N_t, t ≥ 0}, enjoying the following properties

N(t) takes values in the set of non-negative integers, and N(0) = 0.

N(t) has stationary increments, independent over non-overlapping intervals.

N(t) is nontrivial in the sense that, for each t > 0, we have 0 < P(N(t) >0) < 1.

N(t) has only jumps of size 1.