Problem 1.1 In financial mathematics, Markov chains can be used to model the default risk of a company or country (more specifically, the default of a company's or country's liability like a corporate or government bond). Rating agencies like Standard \& Poor's, Moody's, Fitch rate the financial stability of a company and classify them according to different classes. A possible classification may range from 'AAA' for debitors with a very good credit rating to 'CCC' for debitors which are very likely to delay in paying a debt; and 'D' for those debitors who can't satisfy their financial liabilities anymore (in other words, they are default). The yearly credit rating of a company can be modeled as a Markov chain (Xn)n=0,1,2, with state space S={AAA,AA,A,BBB,BB,B,CCC,D} where Xn represents the credit rating class of the company in the n-th year. The transition probabilities are given by (a) Given that today's rating of a company is AAA (i.e., X0=AAA). Compute the probability that the company will not default within the next 8 years. Hint: You can use a computer to answer this question. (b) Determine all recurrent and all transient states