A semigroup approach to nonlinear Lévy processes

Robert Denk, Michael Kupper and Max Nendel

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 (Aλ) λ∈Λ 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 ∂tu=supλ∈ΛAλu. The results are illustrated with several examples.

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Year of publication:	[2019]
Authors:	Denk, Robert ; Kupper, Michael ; Nendel, Max
Publisher:	Bielefeld, Germany : Center for Mathematical Economics (IMW), Bielefeld University
Subject:	Lévy process \| convex expectation space \| Markovian convolution semigroup \| fully nonlinear PDE \| Nisio semigroup \| Stochastischer Prozess \| Stochastic process \| Markov-Kette \| Markov chain \| Nichtlineare Regression \| Nonlinear regression \| Optionspreistheorie \| Option pricing theory \| Finanzmathematik \| Mathematical finance