  We study the homogenization for a class of non-symmetric pure jump Feller
processes. The jump intensity involves periodic and aperiodic constituents, as
well as oscillating and non-oscillating constituents. This means that the noise
can come both from the underlying periodic medium and from external
environments, and is allowed to have different scales. It turns out that the
Feller process converges in distribution, as the scaling parameter goes to
zero, to a L\'evy process. As special cases of our result, some homogenization
problems studied in previous works can be recovered. We also generalize the
approach to the homogenization of symmetric stable-like processes with variable
order. Moreover, we present some numerical experiments to demonstrate the usage
of our homogenization results in the numerical approximation of first exit
times.
