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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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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
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    Citations

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    Cited by:

    1. Criens, David & Niemann, Lars, 2024. "A class of multidimensional nonlinear diffusions with the Feller property," Statistics & Probability Letters, Elsevier, vol. 208(C).
    2. Max Nendel, 2021. "Markov chains under nonlinear expectation," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 474-507, January.
    3. 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.
    4. Martin Larsson & Marvin S. Mueller & Josef Teichmann, 2020. "Stopper-Controller Games embedded in Single-Player Control Problems," Papers 2006.09493, arXiv.org.
    5. 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.
    6. Tolulope Fadina & Ariel Neufeld & Thorsten Schmidt, 2018. "Affine processes under parameter uncertainty," Papers 1806.02912, arXiv.org, revised Mar 2019.
    7. 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.
    8. 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.
    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. 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.
    11. Criens, David & Niemann, Lars, 2024. "Markov selections and Feller properties of nonlinear diffusions," Stochastic Processes and their Applications, Elsevier, vol. 173(C).

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