I have a four-state, discrete time Markov process with time-dependent transition matrices such that after a given time T the matrices become constant. The idea is people in a program leaving the program in a variety of ways. Everyone starts in state 1, and states 2, 3 and 4 are absorbing, but state 4 represents the fairly small percentage of people who are 'lost in the system' - in other words state 4 represents our ignorance of what happens to people rather than a genuine outcome.
I would like to use lumping to put those in state 4 in with those in state 1 and run this as a three-state system, and compare this with the naive approach of running this as a four-state system then apportioning those who are asymptotically in state 4 into states 2 and 3 according to their relative proportions. (in other words, p_2/(p_2 + p_3) of those in state 4 go into state 2 after the system is run to infinite time and similar for state 3)
From some rough scribblings it doesn't seem that these two methods give the same results, so it would be good to get an idea on the error involved. To this end, here's my question:
Can I have pointers to the literature on lumping in Markov chains (or related) that would apply - even roughly - in this example? Or otherwise some words of advice on how to approach this.