Some important properties of any Lèvy processes are:
is an infinite divisible random variable for all
Lèvy -Itô decomposition: can be written as the sum of a diffusion, a continuous Martingale and a pure jump process; i.e:
where , is a standard brownian motion. is defined to be the Poisson random measure of the Lèvy process (the process that counts the jumps): for any Borel in such that then , where ; and is the compensated jump process, which is a martingale.
Some important examples of Lèvy processes include: the Poisson Process, the Compound Poisson process, Brownian Motion, Stable Processes, Subordinators, etc.
Applebaum David (2004). Lèvy Procesess and Stochastic Calculus. Cambridge University Press, Cambrigde, UK.