ABP maximum principles for fully nonlinear integro-differential equations with unbounded inhomogeneous terms

Aleksandrov-Bakelman-Pucci maximum principles are studied for a class of fully nonlinear integro-differential equations of order $\sigma \in [2-{\epsilon }_0,2)$, where ${\epsilon }_0$ is a small constant depending only on given parameters. The goal of this paper is to improve an estimate of Guillen and Schwab (Arch. Ration. Mech. Anal., 206, 2012) in order to avoid the dependence on $L^{\mathrm{\infty }}$ norm of %the to the estimate depending only on $L^n$ norm the inhomogeneous term.