It\^o calculus and jump diffusions for $G$-L\'evy processes

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 $L_G^p(\mathrm{\Omega })$ in terms of their quasi-continuity.