IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2308.03858.html
   My bibliography  Save this paper

Ramifications of generalized Feller theory

Author

Listed:
  • Christa Cuchiero
  • Tonio Mollmann
  • Josef Teichmann

Abstract

Generalized Feller theory provides an important analog to Feller theory beyond locally compact state spaces. This is very useful for solutions of certain stochastic partial differential equations, Markovian lifts of fractional processes, or infinite dimensional affine and polynomial processes which appear prominently in the theory of signature stochastic differential equations. We extend several folklore results related to generalized Feller processes, in particular on their construction and path properties, and provide the often quite sophisticated proofs in full detail. We also introduce the new concept of extended Feller processes and compare them with standard and generalized ones. A key example relates generalized Feller semigroups of algebra homomorphisms via the method of characteristics to transport equations and continuous semiflows on weighted spaces, i.e. a remarkably generic way to treat differential equations on weighted spaces. We also provide a counterexample, which shows that no condition of the basic definition of generalized Feller semigroups can be dropped.

Suggested Citation

  • Christa Cuchiero & Tonio Mollmann & Josef Teichmann, 2023. "Ramifications of generalized Feller theory," Papers 2308.03858, arXiv.org.
    • Handle: RePEc:arx:papers:2308.03858
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2308.03858
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Eduardo Abi Jaber, 2018. "Lifting the Heston model," Papers 1810.04868, arXiv.org, revised Nov 2019.
    2. Eduardo Abi Jaber, 2019. "Lifting the Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 19(12), pages 1995-2013, December.
    3. Eduardo Abi Jaber, 2019. "Lifting the Heston model," Post-Print hal-01890751, HAL.
    4. Christa Cuchiero & Josef Teichmann, 2019. "Markovian lifts of positive semidefinite affine Volterra-type processes," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 407-448, December.
    5. Christa Cuchiero & Josef Teichmann, 2019. "Markovian lifts of positive semidefinite affine Volterra type processes," Papers 1907.01917, arXiv.org, revised Sep 2019.
    6. Imanol Perez Arribas & Cristopher Salvi & Lukasz Szpruch, 2020. "Sig-SDEs model for quantitative finance," Papers 2006.00218, arXiv.org, revised Jun 2020.
    7. Christa Cuchiero & Philipp Schmocker & Josef Teichmann, 2023. "Global universal approximation of functional input maps on weighted spaces," Papers 2306.03303, arXiv.org, revised Feb 2024.
    8. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
    9. Philipp Doersek & Josef Teichmann, 2010. "A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations," Papers 1011.2651, arXiv.org.
    10. Christa Cuchiero & Sara Svaluto-Ferro & Josef Teichmann, 2023. "Signature SDEs from an affine and polynomial perspective," Papers 2302.01362, arXiv.org.
    11. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, vol. 16(4), pages 711-740, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Eduardo Abi Jaber, 2020. "The Laplace transform of the integrated Volterra Wishart process," Working Papers hal-02367200, HAL.
    2. Christa Cuchiero & Sara Svaluto-Ferro, 2021. "Infinite-dimensional polynomial processes," Finance and Stochastics, Springer, vol. 25(2), pages 383-426, April.
    3. Eduardo Abi Jaber, 2019. "The Laplace transform of the integrated Volterra Wishart process," Papers 1911.07719, arXiv.org, revised Jun 2020.
    4. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-02367200, HAL.
    5. Christa Cuchiero & Philipp Schmocker & Josef Teichmann, 2023. "Global universal approximation of functional input maps on weighted spaces," Papers 2306.03303, arXiv.org, revised Feb 2024.
    6. Christa Cuchiero & Guido Gazzani & Janka Moller & Sara Svaluto-Ferro, 2023. "Joint calibration to SPX and VIX options with signature-based models," Papers 2301.13235, arXiv.org.
    7. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Post-Print hal-02367200, HAL.
    8. Ackermann, Julia & Kruse, Thomas & Overbeck, Ludger, 2022. "Inhomogeneous affine Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 250-279.
    9. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Finance and Stochastics, Springer, vol. 26(4), pages 733-769, October.
    10. Christa Cuchiero & Sara Svaluto-Ferro, 2019. "Infinite dimensional polynomial processes," Papers 1911.02614, arXiv.org.
    11. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 309-348, January.
    12. Christa Cuchiero & Sara Svaluto-Ferro & Josef Teichmann, 2023. "Signature SDEs from an affine and polynomial perspective," Papers 2302.01362, arXiv.org.
    13. Matthieu Garcin, 2021. "Forecasting with fractional Brownian motion: a financial perspective," Papers 2105.09140, arXiv.org, revised Sep 2021.
    14. Eduardo Abi Jaber, 2020. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Working Papers hal-02412741, HAL.
    15. Eduardo Abi Jaber & Nathan De Carvalho, 2023. "Reconciling rough volatility with jumps," Papers 2303.07222, arXiv.org.
    16. Siow Woon Jeng & Adem Kiliçman, 2021. "On Multilevel and Control Variate Monte Carlo Methods for Option Pricing under the Rough Heston Model," Mathematics, MDPI, vol. 9(22), pages 1-32, November.
    17. Yiru Xi & Hoi Ying Wong, 2021. "Discrete variance swap in a rough volatility economy," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(10), pages 1640-1654, October.
    18. Etienne Chevalier & Sergio Pulido & Elizabeth Zúñiga, 2021. "American options in the Volterra Heston model," Working Papers hal-03178306, HAL.
    19. Eduardo Abi Jaber, 2021. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-02412741, HAL.
    20. Eduardo Abi Jaber, 2021. "Weak existence and uniqueness for affine stochastic Volterra equations with L1-kernels," Post-Print hal-02412741, HAL.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2308.03858. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.