  The paper considers the integration theory for $G$-L\'evy processes with
finite activity. We introduce the It\^o-L\'evy integrals, give the It\^o
formula for them and establish SDE's, BSDE's and decoupled FBSDE's driven by
$G$-L\'evy processes. In order to develop such a theory, we prove two key
results: the representation of the sublinear expectation associated with a
$G$-L\'evy process and a characterization of random variables in
$L^p_G(\Omega)$ in terms of their quasi-continuity.
