A Semigroup Approach to Nonlinear Lévy Processes

Denk R, Kupper M, Nendel M. A Semigroup Approach to Nonlinear Lévy Processes. Center for Mathematical Economics Working Papers. Vol 610. Bielefeld: Center for Mathematical Economics; 2019.We study the relation between Lévy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear Lévy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators ($A_\lambda$) $_{\lambda\in \Lambda}$ of linear Lévy processes which guarantees the existence of a nonlinear Lévy process such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE $\partial_t u=\sup_{\lambda\in \Lambda} A_\lambda u$. The results are illustrated with several examples