  We study the "periodic homogenization" for a class of nonlocal partial
differential equations of parabolic-type with rapidly oscillating coefficients,
related to stochastic differential equations driven by multiplicative isotropic
$\alpha$-stable L\'evy noise ($1<\alpha<2$) which is nonlinear in the noise
component. Our homogenization method is probabilistic. It turns out that, under
suitable regularity assumptions, the limit of the solutions satisfies a
nonlocal partial differential equation with constant coefficients, which are
associated to a symmetric $\alpha$-stable L\'evy process.
