Lévy processes and continuous - state branching processes : part I

Let us begin by recalling the definition of two familiar processes, a Brownian motion and a Poisson process. A real-valued process B = {B t : t ≥ 0} defined on a probability space (Ω, F , P) is said to be a Brownian motion if the following hold: (i) The paths of B are P-almost surely continuous. (ii) P(B 0 = 0) = 1. (iii) For 0 ≤ s ≤ t, B t − B s is equal in distribution to B t−s. (iv) For 0 ≤ s ≤ t, B t − B s is independent of {B u : u ≤ s}. (v) For each t > 0, B t is equal in distribution to a normal random variable with variance t. A process valued on the non-negative integers N = {N t : t ≥ 0}, defined on a probability space (Ω, F , P), is said to be a Poisson process with intensity λ > 0 if the following hold: (i) The paths of N are P-almost surely right continuous with left limits. (iii) For 0 ≤ s ≤ t, N t − N s is equal in distribution to N t−s. (iv) For 0 ≤ s ≤ t, N t − N s is independent of {N u : u ≤ s}. (v) For each t > 0, N t is equal in distribution to a Poisson random variable with parameter λt. On first encounter, these processes would seem to be considerably different from one another. Firstly, Brownian motion has continuous paths whereas a Poisson process does not. Secondly, a Poisson process is a non-decreasing process 1