Regularity Theory for Fully Nonlinear Integro-Differential Equations

We consider nonlinear integro-differential equations like the ones that arise from stochastic control problems with purely jump Levy processes. We obtain a non-local version of the ABP estimate, Harnack inequality, and interior C-1,C-alpha regularity for general fully nonlinear integro-differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. (C) 2008 Wiley Periodicals, Inc.