Let X = {Xt,t ≥ 0} be a continuous-time homogeneous Markov chain with an intensity matrix Q and state space S = {1, 2, . . . , m}. The sample path of X is observed continuously on a finite interval [0, T ], T < ∞, from which the following statistics are recorded Nij =# i → j transitions of X, Zi =the amount of time X spends in state i ∈ S in total. Using these information, we want to estimate the intensity matrix Q. Let qi = −qii for each i ∈ S . It is known that the likelihood function of the sample path is given by mm L(qij,i, j ∈ S) = Y Y (qij)Nije−qiZi, i=1 j=1, j,i (a) Write down the loglikelihood function. (b) Show that the maximum likelihood estimator bqi j of qi j is given by bq i j = N i j . k,i 0 e−qiuqik pk j(t − u)du. (1) Hint: Recall that q = Pm q . i j=1,j,i ij 1 Zi (c) In a numerical study, a sample path is observed within time window [0, 50]. Con- sider the following realization of sample paths >X [1] 2 1 3 2 1 2 3 2 1 3 2 1 3 >T [1] 0.00 1.43 7.41 10.71 15.03 17.02 25.12 27.12 32.05 33.00 36.60 44.41 52.08 Based on this realizations, estimate the intensity matrix of X. (d) Find the average time the process spends in each state. Compare your findings against the above realizations

Let X = {Xt,t ≥ 0} be a continuous-time homogeneous Markov chain with an intensity matrix Q and state space S = {1, 2, . . . , m}. The sample path of X is observed continuously on a finite interval [0, T ], T < ∞, from which the following statistics are recorded Nij =# i → j transitions of X, Zi =the amount of time X spends in state i ∈ S in total. Using these information, we want to estimate the intensity matrix Q. Let qi = −qii for each i ∈ S . It is known that the likelihood function of the sample path is given by mm L(qij,i, j ∈ S) = Y Y (qij)Nije−qiZi, i=1 j=1, j,i (a) Write down the loglikelihood function. (b) Show that the maximum likelihood estimator bqi j of qi j is given by bq i j = N i j . k,i 0 e−qiuqik pk j(t − u)du. (1) Hint: Recall that q = Pm q . i j=1,j,i ij 1 Zi (c) In a numerical study, a sample path is observed within time window [0, 50]. Con- sider the following realization of sample paths >X [1] 2 1 3 2 1 2 3 2 1 3 2 1 3 >T [1] 0.00 1.43 7.41 10.71 15.03 17.02 25.12 27.12 32.05 33.00 36.60 44.41 52.08 Based on this realizations, estimate the intensity matrix of X. (d) Find the average time the process spends in each state. Compare your findings against the above realizations