IDEAS home Printed from
   My bibliography  Save this article

Stochastic simulations of time series within Weierstrass-Mandelbrot walks


  • R. Kutner
  • F. Switała


In the present work we extend Levy walks to allow the velocity of the walker to vary. We call these extended Levy walks Weierstrass-Mandelbrot walks. This is a generalized model of the Levy walk type which is still able to describe both stationary and non-stationary stochastic time series by treating the initial step of the walker differently. The model was partly motivated by the properties of financial time series and tested on empirical data extracted from the Warsaw stock exchange since it offers an opportunity to study in an unbiased way several features of the stock exchange in its early stages. We extended the continuous-time random walk formalism but the (generalized) waiting-time distribution (WTD) and sojourn probability density still play a fundamental role. We considered a one-dimensional, non-Brownian random walk where the walker moves, in general, with a velocity that assumes a different constant value between the successive turning points, i.e. the velocity is a piecewise constant function. So far the models which have been developed take only one chosen value of this velocity into account and therefore are unable to consider more realistic stochastic time series. Moreover, our model is a kind of Levy walk where we assume a hierarchical, self-similar in the stochastic sense, spatio-temporal representation of WTD and sojourn probability density. The Weierstrass-Mandelbrot walk model makes it possible to analyse both the structure of the Hurst exponent and the power-law behaviour of kurtosis. This structure results from the hierarchical, spatio-temporal coupling between the walker displacement and the corresponding time of the walks. The analysis makes use of both the fractional diffusion and the super-Burnett coefficients. We constructed the diffusion phase diagram which distinguishes regions occupied by classes of different universality. We study only such classes which are characteristic of stationary situations. We proved that even after taking a moving averaging of the stochastic time series which makes results stationary in the sense that they are independent of the beginning moment of the random walk, it is still possible to see the non-Gaussian features of the basic stochastic process. We thus have a model ready for describing data presented, e.g., in the form of moving averages. This operation is often used for stochastic time series, especially financial ones. Based on the hierarchical representation of WTD we introduce an efficient Monte Carlo algorithm which makes a numerical simulation of individual runs of stochastic time series possible; this facilitates the study of empirical stochastic time series.

Suggested Citation

  • R. Kutner & F. Switała, 2003. "Stochastic simulations of time series within Weierstrass-Mandelbrot walks," Quantitative Finance, Taylor & Francis Journals, vol. 3(3), pages 201-211.
  • Handle: RePEc:taf:quantf:v:3:y:2003:i:3:p:201-211
    DOI: 10.1088/1469-7688/3/3/306

    Download full text from publisher

    File URL:
    Download Restriction: Access to full text is restricted to subscribers.

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


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

    Cited by:

    1. Scalas, Enrico & Kaizoji, Taisei & Kirchler, Michael & Huber, Jürgen & Tedeschi, Alessandra, 2006. "Waiting times between orders and trades in double-auction markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 463-471.
    2. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    3. Masoliver, Jaume & Montero, Miquel & Perello, Josep & Weiss, George H., 2006. "The continuous time random walk formalism in financial markets," Journal of Economic Behavior & Organization, Elsevier, vol. 61(4), pages 577-598, December.
    4. Javier Villarroel & Miquel Montero, 2008. "On properties of Continuous-Time Random Walks with Non-Poissonian jump-times," Papers 0812.2148,

    More about this item


    Access and download statistics


    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:taf:quantf:v:3:y:2003:i:3:p:201-211. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Chris Longhurst). General contact details of provider: .

    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.

    We have no references for this item. You can help adding them by using 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.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.