Nonlinear L\'evy Processes and their Characteristics

We develop a general construction for nonlinear L'evy processes with given characteristics. More precisely, given a set $\mathrm{\Theta }$ of L'evy triplets, we construct a sublinear expectation on Skorohod space under which the canonical process has stationary independent increments and a nonlinear generator corresponding to the supremum of all generators of classical L'evy processes with triplets in $\mathrm{\Theta }$. The nonlinear L'evy process yields a tractable model for Knightian uncertainty about the distribution of jumps for which expectations of Markovian functionals can be calculated by means of a partial integro-differential equation.