Levy process

IMHO, the Levy process is just a generalization of the Poisson process. For example, consider $latex X_t$ as the number of arrivals by time $latex t$ as in the Poisson process. We expect the followings will be satisfied

1. $latex X_t =0$ for $latex t = 0$
2. Random variables $latex X_{t_2} – X_{t_1}$ and $latex X_{t_4}-X_{t_3}$ are independent if $latex [t_1,t_2]$ and $latex [t_3,t_4]$ do not overlap
3. $latex Pr(X_{t_2}-X_{t_1}=c)=q_{t_2-t_1}(c)$ for some distribution $latex q$. Meaning that probability distribution only depends on the length of the interval
4. The variable is $latex X_t$ is continuous in probability. That is, for any $latex \epsilon>0$, $latex \lim_{\Delta\rightarrow 0} Pr(|X_{t+\Delta}-X_t|>\epsilon)=0$

They are just natural assumptions that one would take to keep things simple.