IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v462y2016icp1161-1177.html
   My bibliography  Save this article

Fractional randomness

Author

Listed:
  • Tapiero, Charles S.
  • Vallois, Pierre

Abstract

The premise of this paper is that a fractional probability distribution is based on fractional operators and the fractional (Hurst) index used that alters the classical setting of random variables. For example, a random variable defined by its density function might not have a fractional density function defined in its conventional sense. Practically, it implies that a distribution’s granularity defined by a fractional kernel may have properties that differ due to the fractional index used and the fractional calculus applied to define it. The purpose of this paper is to consider an application of fractional calculus to define the fractional density function of a random variable. In addition, we provide and prove a number of results, defining the functional forms of these distributions as well as their existence. In particular, we define fractional probability distributions for increasing and decreasing functions that are right continuous. Examples are used to motivate the usefulness of a statistical approach to fractional calculus and its application to economic and financial problems. In conclusion, this paper is a preliminary attempt to construct statistical fractional models. Due to the breadth and the extent of such problems, this paper may be considered as an initial attempt to do so.

Suggested Citation

  • Tapiero, Charles S. & Vallois, Pierre, 2016. "Fractional randomness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 1161-1177.
    • Handle: RePEc:eee:phsmap:v:462:y:2016:i:c:p:1161-1177
    DOI: 10.1016/j.physa.2016.05.053
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S037843711630245X
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2016.05.053?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. Engle, Robert F & Ito, Takatoshi & Lin, Wen-Ling, 1990. "Meteor Showers or Heat Waves? Heteroskedastic Intra-daily Volatility in the Foreign Exchange Market," Econometrica, Econometric Society, vol. 58(3), pages 525-542, May.
    2. John Barkoulas & Christopher Baum & Mustafa Caglayan, 1999. "Fractional monetary dynamics," Applied Economics, Taylor & Francis Journals, vol. 31(11), pages 1393-1400.
    3. Greene, Myron T. & Fielitz, Bruce D., 1977. "Long-term dependence in common stock returns," Journal of Financial Economics, Elsevier, vol. 4(3), pages 339-349, May.
    4. Dacorogna, Michael M. & Muller, Ulrich A. & Nagler, Robert J. & Olsen, Richard B. & Pictet, Olivier V., 1993. "A geographical model for the daily and weekly seasonal volatility in the foreign exchange market," Journal of International Money and Finance, Elsevier, vol. 12(4), pages 413-438, August.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
    7. Muller, Ulrich A. & Dacorogna, Michel M. & Olsen, Richard B. & Pictet, Olivier V. & Schwarz, Matthias & Morgenegg, Claude, 1990. "Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis," Journal of Banking & Finance, Elsevier, vol. 14(6), pages 1189-1208, December.
    8. Eugene F. Fama, 1963. "Mandelbrot and the Stable Paretian Hypothesis," The Journal of Business, University of Chicago Press, vol. 36, pages 420-420.
    9. Pierre Vallois & Charles Tapiero, 1997. "Range reliability in random walks," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 45(3), pages 325-345, October.
    10. Orsingher, Enzo & Polito, Federico, 2012. "The space-fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 852-858.
    11. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    12. C. W. J. Granger & Roselyne Joyeux, 1980. "An Introduction To Long‐Memory Time Series Models And Fractional Differencing," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 15-29, January.
    13. Vallois, Pierre & Tapiero, Charles S., 2008. "Volatility estimators and the inverse range process in a random volatility random walk and Wiener processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(11), pages 2565-2574.
    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. Tapiero, Charles S. & Vallois, Pierre, 2018. "Fractional Randomness and the Brownian Bridge," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 835-843.
    2. Tapiero, Charles S. & Vallois, Pierre, 2018. "Randomness and fractional stable distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 511(C), pages 54-60.

    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. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 1999. "The Distribution of Exchange Rate Volatility," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-059, New York University, Leonard N. Stern School of Business-.
    2. Baillie, Richard T. & Bollerslev, Tim & Mikkelsen, Hans Ole, 1996. "Fractionally integrated generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 74(1), pages 3-30, September.
    3. Muller, Ulrich A. & Dacorogna, Michel M. & Dave, Rakhal D. & Olsen, Richard B. & Pictet, Olivier V. & von Weizsacker, Jacob E., 1997. "Volatilities of different time resolutions -- Analyzing the dynamics of market components," Journal of Empirical Finance, Elsevier, vol. 4(2-3), pages 213-239, June.
    4. Proelss, Juliane & Schweizer, Denis & Seiler, Volker, 2020. "The economic importance of rare earth elements volatility forecasts," International Review of Financial Analysis, Elsevier, vol. 71(C).
    5. Karuppiah, Jeyanthi & Los, Cornelis A., 2005. "Wavelet multiresolution analysis of high-frequency Asian FX rates, Summer 1997," International Review of Financial Analysis, Elsevier, vol. 14(2), pages 211-246.
    6. Degiannakis, Stavros & Xekalaki, Evdokia, 2004. "Autoregressive Conditional Heteroskedasticity (ARCH) Models: A Review," MPRA Paper 80487, University Library of Munich, Germany.
    7. Segnon, Mawuli & Lux, Thomas, 2013. "Multifractal models in finance: Their origin, properties, and applications," Kiel Working Papers 1860, Kiel Institute for the World Economy (IfW Kiel).
    8. Benoit Mandelbrot & Adlai Fisher & Laurent Calvet, 1997. "A Multifractal Model of Asset Returns," Cowles Foundation Discussion Papers 1164, Cowles Foundation for Research in Economics, Yale University.
    9. Ana Pérez & Esther Ruiz, 2002. "Modelos de memoria larga para series económicas y financieras," Investigaciones Economicas, Fundación SEPI, vol. 26(3), pages 395-445, September.
    10. Bekiros, Stelios D., 2015. "Heuristic learning in intraday trading under uncertainty," Journal of Empirical Finance, Elsevier, vol. 30(C), pages 34-49.
    11. Baviera, Roberto & Pasquini, Michele & Serva, Maurizio & Vergni, Davide & Vulpiani, Angelo, 2001. "Correlations and multi-affinity in high frequency financial datasets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 300(3), pages 551-557.
    12. Malinda & Maya & Jo-Hui & Chen, 2022. "Testing for the Long Memory and Multiple Structural Breaks in Consumer ETFs," Journal of Applied Finance & Banking, SCIENPRESS Ltd, vol. 12(6), pages 1-6.
    13. Bent Jesper Christensen & Morten Ørregaard Nielsen, 2007. "The Effect of Long Memory in Volatility on Stock Market Fluctuations," The Review of Economics and Statistics, MIT Press, vol. 89(4), pages 684-700, November.
    14. Franses,Philip Hans & Dijk,Dick van, 2000. "Non-Linear Time Series Models in Empirical Finance," Cambridge Books, Cambridge University Press, number 9780521779654, January.
    15. Andersen, Torben G. & Bollerslev, Tim, 1997. "Intraday periodicity and volatility persistence in financial markets," Journal of Empirical Finance, Elsevier, vol. 4(2-3), pages 115-158, June.
    16. González-Pla, Francisco & Lovreta, Lidija, 2019. "Persistence in firm’s asset and equity volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    17. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    18. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
    19. Adlai Fisher & Laurent Calvet & Benoit Mandelbrot, 1997. "Multifractality of Deutschemark/US Dollar Exchange Rates," Cowles Foundation Discussion Papers 1166, Cowles Foundation for Research in Economics, Yale University.
    20. Detlef Seese & Christof Weinhardt & Frank Schlottmann (ed.), 2008. "Handbook on Information Technology in Finance," International Handbooks on Information Systems, Springer, number 978-3-540-49487-4, November.

    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:phsmap:v:462:y:2016:i:c:p:1161-1177. 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.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.