Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels : Subcritical Case

Abstract

We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump Lèvy processes. If the index of the operator $σ$ is in $ (1,2)$ (subcritical case), then we obtain a comparison principle, a nonlocal version of the Alexandroff-Backelman-Pucci estimate, a Harnack inequality, a Hölder regularity, and an interior $\rm C^{1,α}$-regularity for fully nonlinear integro-differential equations associated with such a class. Moreover, our estimates remain uniform as the index $σ$ of the operator is getting close to two, so that they can be regarded as a natural extension of regularity results for elliptic partial differential equations.