Consider the model of Section 23.5 but assume there are n possible states. Label them as {1,...,n}. Let Ni be independent Poisson processes with parameters λi, for i = 1,...,n. Assume the state Xt evolves as dXt = n i=1

(i −Xt−)dNit .

(a) Suppose the economy is in state j just before time t (Xt− = j). What is the probability of transiting to state i = j in an instant dt? Does it depend on j?

(b) Let πit denote the conditional probability that the economy is in state i at time t, for i = 1,...,n. Define Xit = 1{Xt=i}. Note that πit is also the conditional probability that the two-state Markov chain Xit is in state 1.

Write down the dynamics (filtering equation) of πit.

(c) Write down a formula for the market price-dividend ratio in terms of the probabilities πit.