in part i) i understood that why process is not markov. it is not markov since following a year with one or more claims: we can move back one level or two levels depending upon the number of claims in the year before the most recent year. so hetre x(t) depends not only on x(t-1) but on x(t-2) also. I am really very confused in ii) part.. how did we split the given levels to form a process y(t) which is markov. i am not able to get it. somebody please elaborate the whole scenario and how we will do the ii) part
You need to consider each of the states in turn and ask yourself: Do I know the onward transition probabilities (ie probabilities of what will happen next) just by knowing the current state, or do I need extra information? If knowing the current state is enough, then that state is Markov and does not need to be split. If you need extra information, then you need to split the state, in order to incorporate the extra information needed in the state itself. So, in the question you're looking at: Level 1: If we make no claim next year, we move up to Level 2. If we make a claim, we must remain in Level 1, regardless of what happened in the year before the most recent one, as we're already on the lowest level. So there's no need to split here, as we know what the possibilities are based on the current state alone. Level 2: If we make no claim next year, we move up to Level 3. If we make a claim, we must move down to Level 1. Again, this is regardless of what happened in the year before the most recent one, as we can move down at most one level from Level 2. So there's no need to split here either. Level 3: If we make no claim next year, we move up to Level 4. That's fine. However, if we make a claim this year but made no claim last year, we move down to Level 2, BUT if we make a claim this year and also made a claim last year, we must move down two levels to Level 1. So knowing we're in Level 3 isn't enough here to decide what can happen next - we also need to know about the number of claims last year. So we split this state up to include information on last year's claims in order to ensure the Markov property holds. Level 4: Have a think about this one yourself.
