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A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations

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  • Philipp Doersek
  • Josef Teichmann

Abstract

We construct normed spaces of real-valued functions with controlled growth on possibly infinite-dimensional state spaces such that semigroups of positive, bounded operators $(P_t)_{t\ge 0}$ thereon with $\lim_{t\to 0+}P_t f(x)=f(x)$ are in fact strongly continuous. This result applies to prove optimal rates of convergence of splitting schemes for stochastic (partial) differential equations with linearly growing characteristics and for sets of functions with controlled growth. Applications are general Da Prato-Zabczyk type equations and the HJM equations from interest rate theory.

Suggested Citation

  • Philipp Doersek & Josef Teichmann, 2010. "A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations," Papers 1011.2651, arXiv.org.
    • Handle: RePEc:arx:papers:1011.2651
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    Citations

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    Cited by:

    1. 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.
    2. Leif Döring & Blanka Horvath & Josef Teichmann, 2017. "Functional Analytic (Ir-)Regularity Properties Of Sabr-Type Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-48, May.
    3. Christa Cuchiero & Josef Teichmann, 2019. "Markovian lifts of positive semidefinite affine Volterra type processes," Papers 1907.01917, arXiv.org, revised Sep 2019.
    4. 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.
    5. Philipp Doersek & Josef Teichmann, 2011. "Efficient simulation and calibration of general HJM models by splitting schemes," Papers 1112.5330, arXiv.org.
    6. Cox, Sonja & Karbach, Sven & Khedher, Asma, 2022. "Affine pure-jump processes on positive Hilbert–Schmidt operators," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 191-229.
    7. Martin Friesen & Sven Karbach, 2024. "Stationary covariance regime for affine stochastic covariance models in Hilbert spaces," Finance and Stochastics, Springer, vol. 28(4), pages 1077-1116, October.
    8. Philipp Doersek & Eskil Hansen, 2012. "High order splitting schemes with complex timesteps and their application in mathematical finance," Papers 1210.5392, arXiv.org.
    9. Blanka Horvath & Oleg Reichmann, 2018. "Dirichlet Forms and Finite Element Methods for the SABR Model," Papers 1801.02719, arXiv.org.
    10. Christa Cuchiero & Tonio Mollmann & Josef Teichmann, 2023. "Ramifications of generalized Feller theory," Papers 2308.03858, arXiv.org.

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