A Markov jump process {Y,t 2 0} has three states {0,1,2} with intensity matrix: -1 0.4 0.6 A = (M;;) = 0.6 -0.6 0 0 0.4 -0.4 (a) Find matrices V , U and Q(t) such that V-'AV =U is an upper-triangular matrix and Q'(t) = Q(t)U with Q(0) =I (you are not required to calculate V and prove the results in this part). [5] (b) Verify AV = VU , Q'(t) = Q(t)U and Q(0) = I . Then prove that the matrix P(t) = VQ(t)V- solves the Kolmogorov forward differential equations P'(t) = P(t) A with P(0) = I. [4] (c) Calculate the first row of the matrix VQ(t) and the first column of V-. Then obtain the transition probability Poo(t) . [3] (d) Calculate the last row of V. Then find lim P(t) (if exists) and verify the limit by R. [3]