I have become stuck on the following problem. Here is the statement:
In a simple birth and death process, the birth rates are
and the death rates are for . The birth and death rates are 0 elsewhere. The parameters and are positive and satisfy . Let . Show that satisfies the following first-order difference equation:
I am thinking that I can use some sort of recurrence relation like where denotes the expectation of for a Markov chain starting at y. However, I haven't been able to get very far with this. One thing that is confusing me is how to relate the expectation of a variable in a chain starting in a particular state with the expectation used in the problem statement (which doesn't seem to specify which state the markov chain started in). Do I have to account for all the possible states that the chain might be in?
Any hints would be greatly appreciated--I'm a little frustrated right now.