A currency speculator makes a profit of more than $5,000 on a trading day with probability p. Let Xn be the length of success run up to and including the nth trading day (n = 0, 1, 2, . . .), where a profit of $5,000 or more is considered a success. For example, if the profits on the consecutive trading days starting from n = 0 are (in thousands of $): 4.5, 7.1, −0.2, 5.7, 6.2, 9.8, 4.9 then X0 = 0, X1 = 1, X2 = 0, X3 = 1, X4 = 2, X5 = 3, X6 = 0.

a) Define the state space and write down the probability transition matrix for the Markov Chain {Xn, n ≥ 0}.
b) For each j ≥ 0, find limn→∞ P(Xn = j).
c) Suppose the trader receives a bonus of $A for every success run of length 3 or more and a bonus of $B (B > A) for every success run of length 6 or more. (The length of the success run is determined after a failure occurs, so, for example, after 7 successful trading days followed by a failure a bonus of $B is awarded and no other bonuses are in order.) Find the bonus payment rate, i.e. the average bonus per day.

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a The state space is 0 1 2 3 4 5 6 7 and the probability transition matrix is as follows ... View the full answer