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Convex monotone semigroups on lattices of continuous functions

Author

Listed:
  • Denk, Robert

    (Center for Mathematical Economics, Bielefeld University)

  • Kupper, Michael

    (Center for Mathematical Economics, Bielefeld University)

  • Nendel, Max

    (Center for Mathematical Economics, Bielefeld University)

Abstract

We consider convex monotone semigroups on a Banach lattice, which is assumed to be a Riesz subspace of a $\sigma$ -Dedekind complete Banach lattice with an additional assumption on the dual space. As typical examples, we consider the space of bounded uniformly continuous functions and the space of continuous functions vanishing at infinity. We show that the domain of the classical generator for convex monotone $C_0$-semigroups, which is defined in terms of the time derivative at 0 w.r.t. the supremum norm, is typically not invariant. We thus propose alternative forms of generators and domains, for which we prove the invariance under the semigroup. As a consequence, we obtain the uniqueness of the semigroup in terms of an extended version of the generator. The results are discussed in several examples related to fully nonlinear partial differential equations, such as uncertain shift semigroups and semigroups related to $G$-heat equations (fully nonlinear versions of the heat equation).

Suggested Citation

  • Denk, Robert & Kupper, Michael & Nendel, Max, 2025. "Convex monotone semigroups on lattices of continuous functions," Center for Mathematical Economics Working Papers 716, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:716
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    File URL: https://pub.uni-bielefeld.de/download/3005039/3005040
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    References listed on IDEAS

    as
    1. Larry G. Epstein & Shaolin Ji, 2013. "Ambiguous Volatility and Asset Pricing in Continuous Time," The Review of Financial Studies, Society for Financial Studies, vol. 26(7), pages 1740-1786.
    2. 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.
    3. Vorbrink, Jörg, 2014. "Financial markets with volatility uncertainty," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 64-78.
    4. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    5. 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.
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