IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v119y2009i6p1932-1951.html

First jump approximation of a Lévy-driven SDE and an application to multivariate ECOGARCH processes

Author

Listed:
  • Stelzer, Robert

Abstract

The first jump approximation of a pure jump Lévy process, which converges to the Lévy process in the Skorokhod topology in probability, is generalised to a multivariate setting and an infinite time horizon. It is shown that it can generally be used to obtain "first jump approximations" of Lévy-driven stochastic differential equations, by establishing that it has uniformly controlled variations. Applying this general result to multivariate exponential continuous time GARCH processes of order (1, 1), it is shown that there exists a sequence of piecewise constant processes determined by multivariate exponential GARCH(1, 1) processes in discrete time which converge in probability in the Skorokhod topology to the continuous time process.

Suggested Citation

  • Stelzer, Robert, 2009. "First jump approximation of a Lévy-driven SDE and an application to multivariate ECOGARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 119(6), pages 1932-1951, June.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:6:p:1932-1951
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(08)00146-4
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Rubenthaler, Sylvain, 2003. "Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 103(2), pages 311-349, February.
    2. Szimayer, Alex & Maller, Ross A., 2007. "Finite approximation schemes for Lévy processes, and their application to optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1422-1447, October.
    3. Ross A. Maller & Gernot Muller & Alex Szimayer, 2008. "GARCH modelling in continuous time for irregularly spaced time series data," Papers 0805.2096, arXiv.org.
    4. Kawakatsu, Hiroyuki, 2006. "Matrix exponential GARCH," Journal of Econometrics, Elsevier, vol. 134(1), pages 95-128, September.
    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. Müller, Gernot & Durand, Robert B. & Maller, Ross A., 2011. "The risk-return tradeoff: A COGARCH analysis of Merton's hypothesis," Journal of Empirical Finance, Elsevier, vol. 18(2), pages 306-320, March.

    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. Müller, Gernot & Durand, Robert B. & Maller, Ross A., 2011. "The risk-return tradeoff: A COGARCH analysis of Merton's hypothesis," Journal of Empirical Finance, Elsevier, vol. 18(2), pages 306-320, March.
    2. Francesco Bianchi & Lorenzo Mercuri & Edit Rroji, 2022. "Portfolio Selection with Irregular Time Grids: an example using an ICA-COGARCH(1, 1) approach," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 36(1), pages 57-85, March.
    3. Han, Chulwoo & Park, Frank C., 2022. "A geometric framework for covariance dynamics," Journal of Banking & Finance, Elsevier, vol. 134(C).
    4. Mei Xing, 2017. "Existence Conditions of Super-Replication Cost in a Multinomial Model," Journal of Mathematics Research, Canadian Center of Science and Education, vol. 9(4), pages 185-195, August.
    5. Taufer, Emanuele & Leonenko, Nikolai, 2009. "Simulation of Lvy-driven Ornstein-Uhlenbeck processes with given marginal distribution," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2427-2437, April.
    6. Sucarrat, Genaro & Grønneberg, Steffen & Escribano, Alvaro, 2016. "Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 582-594.
    7. Manabu Asai & Michael McAleer, 2009. "Dynamic Conditional Correlations for Asymmetric Processes," CARF F-Series CARF-F-168, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    8. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, vol. 13(4), pages 471-500, September.
    9. Siddhartha Chib & Yasuhiro Omori & Manabu Asai, 2009. "Multivariate Stochastic Volatility," Springer Books, in: Thomas Mikosch & Jens-Peter Kreiß & Richard A. Davis & Torben Gustav Andersen (ed.), Handbook of Financial Time Series, chapter 16, pages 365-400, Springer.
    10. Barrera, Carlos, 2010. "¿Respuesta asimétrica de precios domésticos de combustibles ante choques en el WTI?," Working Papers 2010-016, Banco Central de Reserva del Perú.
    11. Tim Bollerslev, 2008. "Glossary to ARCH (GARCH)," CREATES Research Papers 2008-49, Department of Economics and Business Economics, Aarhus University.
    12. Panloup, Fabien, 2008. "Computation of the invariant measure for a Lévy driven SDE: Rate of convergence," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1351-1384, August.
    13. Carlos R. Barrera Chaupis, 2018. "Inventory Adjustments to Demand Shocks under Flexible Specifications," Monetaria, Centro de Estudios Monetarios Latinoamericanos, CEMLA, vol. 0(1), pages 149-201, january-j.
    14. Magnus, Jan R. & Pijls, Henk G.J. & Sentana, Enrique, 2021. "The Jacobian of the exponential function," Journal of Economic Dynamics and Control, Elsevier, vol. 127(C).
    15. Hafner, Christian M. & Wang, Linqi, 2023. "A dynamic conditional score model for the log correlation matrix," Journal of Econometrics, Elsevier, vol. 237(2).
    16. Dereich, Steffen & Heidenreich, Felix, 2011. "A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1565-1587, July.
    17. Barrera, Carlos R., 2011. "Impacto amplificador del ajuste de inventarios ante choques de demanda según especificaciones flexibles," Working Papers 2011-009, Banco Central de Reserva del Perú.
    18. Asai, Manabu & McAleer, Michael, 2015. "Forecasting co-volatilities via factor models with asymmetry and long memory in realized covariance," Journal of Econometrics, Elsevier, vol. 189(2), pages 251-262.
    19. Duan, Wei-Long & Fang, Hui & Zeng, Chunhua, 2019. "Second-order algorithm for simulating stochastic differential equations with white noises," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 525(C), pages 491-497.
    20. P. Brockwell, 2014. "Recent results in the theory and applications of CARMA processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 647-685, August.

    More about this item

    Keywords

    ;

    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:eee:spapps:v:119:y:2009:i:6:p:1932-1951. 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.