IDEAS home Printed from
MyIDEAS: Log in (now much improved!) to save this article

Stochastic simulations of time series within Weierstrass-Mandelbrot walks

Listed author(s):
  • 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.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

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

As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

Article provided by Taylor & Francis Journals in its journal Quantitative Finance.

Volume (Year): 3 (2003)
Issue (Month): 3 ()
Pages: 201-211

in new window

Handle: RePEc:taf:quantf:v:3:y:2003:i:3:p:201-211
DOI: 10.1088/1469-7688/3/3/306
Contact details of provider: Web page:

Order Information: Web:

No references listed on IDEAS
You can help add them by filling out this form.

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

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)

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 references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link 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 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.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.