  This chapter is an attempt to present a mathematical theory of compound
fractional Poisson processes. The chapter begins with the characterization of a
well-known L\'evy process: The compound Poisson process. The semi-Markov
extension of the compound Poisson process naturally leads to the compound
fractional Poisson process, where the Poisson counting process is replaced by
the Mittag-Leffler counting process also known as fractional Poisson process.
This process is no longer Markovian and L\'evy. However, several analytical
results are available and some of them are discussed here. The functional limit
of the compound Poisson process is an $\alpha$-stable L\'evy process, whereas
in the case of the compound fractional Poisson process, one gets an
$\alpha$-stable L\'evy process subordinated to the fractional Poisson process.

Probability;Statistical Finance