### 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).

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