Stochastic Integration with respect to Counting Processes

This applet puts the last two applets together: you can change both the speed and the jump size.

The result is a stochastic integration with respect to a time changed Poisson process.

The initial m(s)=1, speed(s)=1, so 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).

For whatever speed and jump size you may click, the resulting red line is always the integration of m(s)speed(s)ds from 0 to t. The black line is integration of m(s) d N( speed(s)) from 0 to t.

It is also true that the difference between the black and red plot is always a martingale.

How to use this applet:

Click on "Start" to begin. Click "Increase jump size" or "Decrease jump size" to see the effects of m(t); click on "Faster" or "Slower" to see the effects of speed(t).

To simulate the Nelson-Aalen estimator based on a sample, you would keep m(s)= 1/R(s) where R(s) is the number of subjects at risk at time s; and speed(s) = sum h_i (s) where the summation is over all subjects that are still at risk at time s. (h_i(s) is the hazard function of the ith subject at time s).

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 install the java virtual machine (which is freely downloadable.

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