  In this paper we establish periodic homogenization for
Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of
integro-differential type. We consider the case when the fractional diffusion
has the same order as the drift term, and is weakly elliptic. The outcome of
the paper is twofold. One one hand, we provide Lipschitz regularity results for
weakly elliptic non-local HJB, extending the results previously obtained in
[8]. On the other hand, we establish a convergence result, based on half
relaxed limits and a comparison principle for the effective problem. The latter
strongly relies on the regularity and the ellipticity properties of the
effective Hamiltonian, for which a fine Lipschitz estimate of the corrector
plays a crucial role.
