  We consider a class of fully nonlinear integro-differential operators where
the nonlocal integral has two components: the non-degenerate one corresponds to
the $\alpha$-stable operator and the second one (possibly degenerate)
corresponds to a class of \textit{lower order} L\'evy measures. Such operators
do not have a global scaling property. We establish H\"{o}lder regularity,
Harnack inequality and boundary Harnack property of solutions of these
operators.
