IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v125y2015i8p2989-3022.html
   My bibliography  Save this article

A generalised Itō formula for Lévy-driven Volterra processes

Author

Listed:
  • Bender, Christian
  • Knobloch, Robert
  • Oberacker, Philip

Abstract

We derive a generalised Itō formula for stochastic processes which are constructed by a convolution of a deterministic kernel with a centred Lévy process. This formula has a unifying character in the sense that it contains the classical Itō formula for Lévy processes as well as recent change-of-variable formulas for Gaussian processes such as fractional Brownian motion as special cases. Our result also covers fractional Lévy processes (with Mandelbrot–Van Ness kernel) and a wide class of related processes for which such a generalised Itō formula has not yet been available in the literature.

Suggested Citation

  • Bender, Christian & Knobloch, Robert & Oberacker, Philip, 2015. "A generalised Itō formula for Lévy-driven Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 2989-3022.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:8:p:2989-3022
    DOI: 10.1016/j.spa.2015.02.009
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915000599
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2015.02.009?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Barndorff-Nielsen, Ole E. & Benth, Fred Espen & Pedersen, Jan & Veraart, Almut E.D., 2014. "On stochastic integration for volatility modulated Lévy-driven Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 812-847.
    2. Christian Bender & Alexander Lindner & Markus Schicks, 2012. "Finite Variation of Fractional Lévy Processes," Journal of Theoretical Probability, Springer, vol. 25(2), pages 594-612, June.
    3. Neuman, Eyal, 2014. "The multifractal nature of Volterra–Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 3121-3145.
    4. Bender, Christian, 2014. "Backward SDEs driven by Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2892-2916.
    5. Basse, Andreas & Pedersen, Jan, 2009. "Lévy driven moving averages and semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2970-2991, September.
    6. Bender, Christian, 2003. "An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 81-106, March.
    7. Ole E. Barndorff-Nielsen & Fred Espen Benth & Almut E. D. Veraart, 2013. "Modelling energy spot prices by volatility modulated L\'{e}vy-driven Volterra processes," Papers 1307.6332, arXiv.org.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Pakkanen, Mikko S. & Sottinen, Tommi & Yazigi, Adil, 2017. "On the conditional small ball property of multivariate Lévy-driven moving average processes," Stochastic Processes and their Applications, Elsevier, vol. 127(3), pages 749-782.

    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. Almut E. D. Veraart & Luitgard A. M. Veraart, 2013. "Risk premia in energy markets," CREATES Research Papers 2013-02, Department of Economics and Business Economics, Aarhus University.
    2. Mark Podolskij, 2014. "Ambit fields: survey and new challenges," CREATES Research Papers 2014-51, Department of Economics and Business Economics, Aarhus University.
    3. Mark Podolskij & Nopporn Thamrongrat, 2015. "A weak limit theorem for numerical approximation of Brownian semi-stationary processes," CREATES Research Papers 2015-53, Department of Economics and Business Economics, Aarhus University.
    4. Barndorff-Nielsen, Ole E. & Benth, Fred Espen & Pedersen, Jan & Veraart, Almut E.D., 2014. "On stochastic integration for volatility modulated Lévy-driven Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 812-847.
    5. Ole E. Barndorff-Nielsen, 2016. "Assessing Gamma kernels and BSS/LSS processes," CREATES Research Papers 2016-09, Department of Economics and Business Economics, Aarhus University.
    6. Mikkel Bennedsen & Asger Lunde & Mikko S. Pakkanen, 2017. "Decoupling the short- and long-term behavior of stochastic volatility," CREATES Research Papers 2017-26, Department of Economics and Business Economics, Aarhus University.
    7. Ole E. Barndorff-Nielsen & Mikko S. Pakkanen & Jürgen Schmiegel, 2013. "Assessing Relative Volatility/Intermittency/Energy Dissipation," CREATES Research Papers 2013-15, Department of Economics and Business Economics, Aarhus University.
    8. Axel A. Araneda, 2019. "The fractional and mixed-fractional CEV model," Papers 1903.05747, arXiv.org, revised Jun 2019.
    9. Robert Elliott & Leunglung Chan, 2004. "Perpetual American options with fractional Brownian motion," Quantitative Finance, Taylor & Francis Journals, vol. 4(2), pages 123-128.
    10. Axel A. Araneda, 2021. "Price modelling under generalized fractional Brownian motion," Papers 2108.12042, arXiv.org, revised Nov 2023.
    11. Feng, Chengxiao & Tan, Jie & Jiang, Zhenyu & Chen, Shuang, 2020. "A generalized European option pricing model with risk management," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    12. Sauri, Orimar & Veraart, Almut E.D., 2017. "On the class of distributions of subordinated Lévy processes and bases," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 475-496.
    13. Ladokhin, Sergiy & Borovkova, Svetlana, 2021. "Three-factor commodity forward curve model and its joint P and Q dynamics," Energy Economics, Elsevier, vol. 101(C).
    14. Andrés Cárdenas & Sergio Pulido & Rafael Serrano, 2022. "Existence of optimal controls for stochastic Volterra equations," Working Papers hal-03720342, HAL.
    15. Harms, Philipp & Stefanovits, David, 2019. "Affine representations of fractional processes with applications in mathematical finance," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1185-1228.
    16. Eduardo Rossi & Paolo Santucci de Magistris, 2018. "Indirect inference with time series observed with error," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 33(6), pages 874-897, September.
    17. Russo, Francesco & Tudor, Ciprian A., 2006. "On bifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 116(5), pages 830-856, May.
    18. Fred Espen Benth & Hanna Zdanowicz, 2016. "Pricing And Hedging Of Energy Spread Options And Volatility Modulated Volterra Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(01), pages 1-22, February.
    19. Yan, Litan, 2004. "Maximal inequalities for the iterated fractional integrals," Statistics & Probability Letters, Elsevier, vol. 69(1), pages 69-79, August.
    20. Douissi, Soukaina & Wen, Jiaqiang & Shi, Yufeng, 2019. "Mean-field anticipated BSDEs driven by fractional Brownian motion and related stochastic control problem," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 282-298.

    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:eee:spapps:v:125:y:2015:i:8:p:2989-3022. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.