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A semigroup approach to nonlinear Lévy processes

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2020

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https://doi.org/10.1016/j.spa.2019.05.009
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http://arxiv.org/abs/1710.08130

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Stochastic Processes and their Applications. Elsevier. 2020, 130(3), pp. 1616-1642. ISSN 0304-4149. eISSN 1879-209X. Available under: doi: 10.1016/j.spa.2019.05.009

Zusammenfassung

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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Fachgebiet (DDC)
510 Mathematik

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Lévy process; Convex expectation space; Markovian convolution semigroup; Fully nonlinear PDE; Nisio semigroup

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ISO 690DENK, Robert, Michael KUPPER, Max NENDEL, 2020. A semigroup approach to nonlinear Lévy processes. In: Stochastic Processes and their Applications. Elsevier. 2020, 130(3), pp. 1616-1642. ISSN 0304-4149. eISSN 1879-209X. Available under: doi: 10.1016/j.spa.2019.05.009
BibTex
@article{Denk2020semig-40474.2,
  year={2020},
  doi={10.1016/j.spa.2019.05.009},
  title={A semigroup approach to nonlinear Lévy processes},
  number={3},
  volume={130},
  issn={0304-4149},
  journal={Stochastic Processes and their Applications},
  pages={1616--1642},
  author={Denk, Robert and Kupper, Michael and Nendel, Max}
}
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