  We introduce a new class of fully nonlinear integro-differential operators
with possible nonsymmetric kernels, which includes the ones that arise from
stochastic control problems with purely jump L\`evy processes. If the index of
the operator $\sigma$ is in $ (1,2)$ (subcritical case), then we obtain a
comparison principle, a nonlocal version of the Alexandroff-Backelman-Pucci
estimate, a Harnack inequality, a H\"older regularity, and an interior $\rm
C^{1,\alpha}$-regularity for fully nonlinear integro-differential equations
associated with such a class. Moreover, our estimates remain uniform as the
index $\sigma$ of the operator is getting close to two, so that they can be
regarded as a natural extension of regularity results for elliptic partial
differential equations.
