Aleksandrov-Bakelman-Pucci maximum principles for a class of uniformly elliptic and parabolic integro-PDE

We prove generalized Aleksandrov-Bakelman-Pucci maximum principles for elliptic and parabolic integro-PDEs of Hamilton-Jacobi-Bellman-Isaacs types, whose PDE parts are either uniformly elliptic or uniformly parabolic. The proofs of these results are based on the classical Aleksandrov-Bakelman-Pucci maximum principles for the elliptic and parabolic PDEs and an iteration procedure using solutions of Pucci extremal equations. We also provide proofs of nonlocal versions of the classical Aleksandrov-Bakelman-Pucci maximum principles for elliptic and parabolic integro-PDEs.