Consider the second version of the stock market model presented as an example in Sec. 16.2. Whether the stock goes up tomorrow depends upon whether it increased today and yesterday.

If the stock increased today and yesterday, it will increase tomorrow with probability 1. If the stock increased today and decreased yesterday, it will increase tomorrow with probability 2. If the stock decreased today and increased yesterday, it will increase tomorrow with probability 3. Finally, if the stock decreased today and yesterday, it will increase tomorrow with probability 4.

(a) Construct the (one-step) transition matrix of the Markov chain.

(b) Explain why the states used for this Markov chain cause the mathematical definition of the Markovian property to hold even though what happens in the future (tomorrow) depends upon what happened in the past (yesterday) as well as the present (today).