IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v24y2011i3d10.1007_s10959-010-0320-9.html
   My bibliography  Save this article

Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations

Author

Listed:
  • Kei Kobayashi

    (Tufts University)

Abstract

It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the Itô formula is derived. When a standard Brownian motion is the original semimartingale, classical Itô stochastic differential equations driven by the Brownian motion with drift extend to a larger class of stochastic differential equations involving a time-change with continuous paths. A form of the general solution of linear equations in this new class is established, followed by consideration of some examples analogous to the classical equations. Through these examples, each coefficient of the stochastic differential equations in the new class is given meaning. The new feature is the coexistence of a usual drift term along with a term related to the time-change.

Suggested Citation

  • Kei Kobayashi, 2011. "Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 24(3), pages 789-820, September.
  • Handle: RePEc:spr:jotpro:v:24:y:2011:i:3:d:10.1007_s10959-010-0320-9
    DOI: 10.1007/s10959-010-0320-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-010-0320-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10959-010-0320-9?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. Bondesson, Lennart & Kristiansen, Gundorph K. & Steutel, Fred W., 1996. "Infinite divisibility of random variables and their integer parts," Statistics & Probability Letters, Elsevier, vol. 28(3), pages 271-278, July.
    2. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2008. "Triangular array limits for continuous time random walks," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1606-1633, 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. Alessandro Gregorio & Francesco Iafrate, 2024. "Path Dynamics of Time-Changed Lévy Processes: A Martingale Approach," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3246-3280, November.
    2. Bareche, Aîcha & Bibi, Abdelouahab, 2023. "On inverse-Gamma distribution delayed by Poisson process," Statistics & Probability Letters, Elsevier, vol. 195(C).

    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. Mijena, Jebessa B. & Nane, Erkan, 2014. "Correlation structure of time-changed Pearson diffusions," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 68-77.
    2. Fernandez-Anaya, G. & Valdes-Parada, F.J. & Alvarez-Ramirez, J., 2011. "On generalized fractional Cattaneo’s equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4198-4202.
    3. P. Escalona & F. Ordóñez & I. Kauak, 2017. "Critical level rationing in inventory systems with continuously distributed demand," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 39(1), pages 273-301, January.
    4. Magdziarz, M. & Scheffler, H.P. & Straka, P. & Zebrowski, P., 2015. "Limit theorems and governing equations for Lévy walks," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4021-4038.
    5. Christoph, Gerd & Schreiber, Karina, 2000. "Scaled Sibuya distribution and discrete self-decomposability," Statistics & Probability Letters, Elsevier, vol. 48(2), pages 181-187, June.
    6. Kumar, A. & Vellaisamy, P., 2015. "Inverse tempered stable subordinators," Statistics & Probability Letters, Elsevier, vol. 103(C), pages 134-141.
    7. D’Ovidio, Mirko, 2012. "From Sturm–Liouville problems to fractional and anomalous diffusions," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3513-3544.
    8. Giacomo Ascione & Nikolai Leonenko & Enrica Pirozzi, 2022. "Non-local Solvable Birth–Death Processes," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1284-1323, June.
    9. Kerger, Phillip & Kobayashi, Kei, 2020. "Parameter estimation for one-sided heavy-tailed distributions," Statistics & Probability Letters, Elsevier, vol. 164(C).
    10. Kondratiev, Yuri & da Silva, José L., 2023. "Compound Poisson processes: Potentials, Green measures and random times," Statistics & Probability Letters, Elsevier, vol. 197(C).
    11. Meerschaert, Mark M. & Nane, Erkan & Xiao, Yimin, 2013. "Fractal dimension results for continuous time random walks," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1083-1093.
    12. Baeumer, B. & Meerschaert, M.M., 2007. "Fractional diffusion with two time scales," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 237-251.
    13. Gupta, Neha & Kumar, Arun, 2022. "Inverse tempered stable subordinators and related processes with Mellin transform," Statistics & Probability Letters, Elsevier, vol. 186(C).
    14. Beghin, Luisa & Macci, Claudio & Ricciuti, Costantino, 2020. "Random time-change with inverses of multivariate subordinators: Governing equations and fractional dynamics," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6364-6387.
    15. Meerschaert, Mark M. & Toaldo, Bruno, 2019. "Relaxation patterns and semi-Markov dynamics," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2850-2879.
    16. Lennart Bondesson, 2002. "On the Lévy Measure of the Lognormal and the LogCauchy Distributions," Methodology and Computing in Applied Probability, Springer, vol. 4(3), pages 243-256, September.
    17. Torricelli, Lorenzo, 2020. "Trade duration risk in subdiffusive financial models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 541(C).
    18. Kumar, A. & Wyłomańska, A. & Połoczański, R. & Sundar, S., 2017. "Fractional Brownian motion time-changed by gamma and inverse gamma process," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 648-667.
    19. Choe, Geon Ho & Lee, Dong Min, 2016. "Numerical computation of hitting time distributions of increasing Lévy processes," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 289-294.
    20. A. Kumar & J. Gajda & A. Wyłomańska & R. Połoczański, 2019. "Fractional Brownian Motion Delayed by Tempered and Inverse Tempered Stable Subordinators," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 185-202, March.

    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:spr:jotpro:v:24:y:2011:i:3:d:10.1007_s10959-010-0320-9. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.