9769

Stable Lévy Process

This Demonstration shows the path of a symmetric stable Lévy process. Such a process has the property of "self-similarity" with Hurst exponent (where is the index of stability) and independent increments (unlike fractional Brownian motion, whose increments are not independent except in the case of Hurst exponent ). For , the stable Lévy process coincides with ordinary Brownian motion. In this Demonstration, you can vary the parameter between 0 and 2 and the vertical range of the plot to zoom in and out.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

One of the most remarkable properties of Brownian motion is self-similarity: for all and all , the random variable has the same distribution as . A strictly stable Lévy process can be viewed as a generalization of Brownian motion, and has the property for all , , has the same distribution as , for some index of stability , where . For each index of stability , the distributions are stable distributions . The probability density functions of this distribution are not known in explicit form except in special cases. In this Demonstration we consider the symmetric case, for which . In this case the characteristic function is given by , which shows that for we get the Cauchy distribution and for the normal distribution.
A symmetric -stable process can be represented as a combination of a (compound) Poisson process and a Brownian motion. For small values of we see that the process is dominated by big jumps. For medium values (e.g., , i.e., Cauchy process) we get both small and large jumps. For close to 2 we get Brownian motion with occasional jumps.
-stable processes for have infinite variance, which makes them somewhat inconvenient. Nevertheless, they are important in physics, biology, meteorology, and have been used in option pricing in finance.
G. Samrodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes, New York: Chapman and Hall, 1994.
A. N. Shiryaev, Essentials of Stochastic Finance, River Edge, New Jersey: World Scientific Publishing Co., 1999.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+