Regularity for solutions of non local parabolic equations

We study the regularity of solutions of parabolic fully nonlinear nonlocal equations. We prove C (alpha) regularity in space and time and, under different assumptions on the kernels, C (1,alpha) in space for translation invariant equations. The proofs rely on a weak parabolic ABP and the classic ideas of Tso (Commun. Partial Diff. Equ. 10(5):543-553, 1985) and Wang (Commun. Pure Appl. Math. 45(1), 27-76, 1992). Our results remain uniform as sigma -> 2 allowing us to recover most of the regularity results found in Tso (Commun. Partial Diff. Equ. 10(5):543-553, 1985).