Non-homogenous Poisson Process

A counting process denoted by  is said to be a non-stationary or non-homogeneous Poisson process with intensity function ,  if it has the following four properties which are:

• : This means the number of occurrences at time , i.e., when the process has just started is zero.
  
  
•  has independent increments.
  
  
• : Which means that the probability of the number of events in the time interval  being exactly equal to 1 is equal to the product of the rate of the process (which is dependent on time, ) and the time interval plus some incremental function of time interval, i.e., .
  
  
• : The fourth property denotes that in case we are interested to find the probability that the number of events in the time interval  is equal to 2 or more, then that probability becomes zero as the time interval shrinks or is made smaller and smaller. For the benefit of the reader we like to mention that a function  is said to be  if we have