9711

The Poisson Process

This Demonstration shows sample trajectories of a Poisson process—a fundamental example of a stochastic process with discontinuous trajectories, which is used as a building block for constructing jump processes that play an important role in modern financial modeling. You can vary the intensity of the jumps, the duration, whether the trajectories should be connected, and whether an ordinary Poisson process or a "compensated" one should be displayed.

SNAPSHOTS

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

DETAILS

The Poisson process is one of the two most fundamental stochastic processes in continuous time financial modelling, the other being Brownian motion, with which it shares a number of common properties (both are examples of a Levy process). In this Demonstration two kinds of Poisson processes are shown: an ordinary one and one that is "compensated". The compensated Poisson process is a martingale with expected value 0. As the intensity of jumps increases, the compensated Poisson process approximates a Brownian motion (this follows from the Donsker invariance principle). Making the plot joined and choosing a high value for the intensity makes the trajectories resemble those in standard simulations of Brownian motion.
R. Cont, P. Tankov, Financial Modelling with Jump Processes, Boca Raton, London, New York, Washington, D.C: Chapman & Hall, 2004.
    • 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+