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A semigroup approach to nonlinear Lévy processes

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  • Denk, Robert
  • Kupper, Michael
  • Nendel, Max

Abstract

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.

Suggested Citation

  • Denk, Robert & Kupper, Michael & Nendel, Max, 2020. "A semigroup approach to nonlinear Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1616-1642.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:3:p:1616-1642
    DOI: 10.1016/j.spa.2019.05.009
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    1. Peng, Shige, 2008. "Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2223-2253, December.
    2. Dolinsky, Yan & Nutz, Marcel & Soner, H. Mete, 2012. "Weak approximation of G-expectations," Stochastic Processes and their Applications, Elsevier, vol. 122(2), pages 664-675.
    3. Neufeld, Ariel & Nutz, Marcel, 2014. "Measurability of semimartingale characteristics with respect to the probability law," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3819-3845.
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    Cited by:

    1. Tolulope Fadina & Ariel Neufeld & Thorsten Schmidt, 2018. "Affine processes under parameter uncertainty," Papers 1806.02912, arXiv.org, revised Mar 2019.
    2. Criens, David & Niemann, Lars, 2024. "A class of multidimensional nonlinear diffusions with the Feller property," Statistics & Probability Letters, Elsevier, vol. 208(C).
    3. Criens, David & Niemann, Lars, 2024. "Markov selections and Feller properties of nonlinear diffusions," Stochastic Processes and their Applications, Elsevier, vol. 173(C).
    4. Lianzi Jiang & Gechun Liang, 2024. "A Robust $$\alpha $$ α -Stable Central Limit Theorem Under Sublinear Expectation without Integrability Condition," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2394-2424, September.
    5. Martin Larsson & Marvin S. Mueller & Josef Teichmann, 2020. "Stopper-Controller Games embedded in Single-Player Control Problems," Papers 2006.09493, arXiv.org.
    6. Nendel, Max & Röckner, Michael, 2019. "Upper Envelopes of Families of Feller Semigroups and Viscosity Solutions to a Class of Nonlinear Cauchy Problems," Center for Mathematical Economics Working Papers 618, Center for Mathematical Economics, Bielefeld University.
    7. Denk, Robert & Kupper, Michael & Nendel, Max, 2019. "Convex Semigroups on Banach Lattices," Center for Mathematical Economics Working Papers 622, Center for Mathematical Economics, Bielefeld University.
    8. Max Nendel, 2021. "Markov chains under nonlinear expectation," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 474-507, January.
    9. Daniel Bartl & Stephan Eckstein & Michael Kupper, 2020. "Limits of random walks with distributionally robust transition probabilities," Papers 2007.08815, arXiv.org, revised Apr 2021.
    10. Changhong Guo & Shaomei Fang & Yong He, 2023. "A Generalized Stochastic Process: Fractional G-Brownian Motion," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-34, March.
    11. Fuhrmann, Sven & Kupper, Michael & Nendel, Max, 2021. "Wasserstein Perturbations of Markovian Transition Semigroups," Center for Mathematical Economics Working Papers 649, Center for Mathematical Economics, Bielefeld University.

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