  In this paper we are interested on the well-posedness of Dirichlet problems
associated to integro-differential elliptic operators of order $\alpha < 1$ in
a bounded smooth domain $\Omega$ . The main difficulty arises because of losses
of the boundary condition for sub and supersolutions due to the lower diffusive
effect of the elliptic operator compared with the drift term. We consider the
notion of viscosity solution with generalized boundary conditions, concluding
strong comparison principles in $\bar{\Omega}$ under rather general assumptions
over the drift term. As a consequence, existence and uniqueness of solutions in
$C(\bar{\Omega})$ is obtained via Perron's method.
