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We prove an interior regularity result for solutions of a purelyintegro-differential Bellman equation. This regularity is enough for thesolutions to be understood in the classical sense. If we let the order of theequation approach two, we recover the theorem of Evans and Krylov about theregularity of solutions to concave uniformly elliptic partial differentialequations.

The Evans-Krylov theorem for non local fully non linear equations