Stochastic Integration with respect to Counting Processes

Counting processes can only have jumps of size one. If the jump sizes can vary, then it will be more useful to model real life events. (For example FED jumps interest rate by 25 base points, or 50 base points, or lower it by 25 points, etc).

This can be achieved by the stochastic integration with respect to a counting process. In the integration you may specify a magnitude function m(t), so that IF there is a jump at t, the size of the jump will be m(t). If m(t) is always = 1 then you get back the original counting process.

In the following plot, we illustrate the integration with respect to a Poisson process. So the time of the jumps always happen according to a Poisson process, but the magnitude of the jumps are controlled by you. Click on "Increase jump size" or "Decrease jump size" and watch it change as it printed on the top of the plot as "Current jump size". The jump size may be changed in anyway you like. The jump size can be negative.

If you do nothing other than click the "Start" button, then the plot shows a standard Poisson process N(t) (black) and its cumulative intensity t (red). Of course, the more interesting thing about this applet is that it allows you to change the jump size.

For whatever jump size you may click, the resulting red line is always the integration of m(s)ds from 0 to t. It is also true that the difference between the black and red plot is a martingale.

How to use this applet:

Click on "Start" to begin. Click "Increase jump size" or "Decrease jump size" to see the effects.

The Java applet should have appeared here. Since it has not, this means (a) your browser is not capable of viewing Java applets (In which case you should consider downloading a new browser) or (b) you need to check your browser settings to enable Java.

Written by Mai Zhou Copyright © 2005 Mai Zhou. All rights reserved.