An insurance company is using a Healthy-Sick-Dead (HSD) model, which is a Markov jump proce to monitor the current state of its Income Protection policyholders, with the following transition rate diagram of annual rates as functions of age t : 0.001 + 0.0003 t H: Healthy S: Sick 0.5 -0.005 t 0.0004 + 0.0002 t 0.0009 + 0.0007t D: Dead (i) Code a function to create a generator matrix A(t) that includes the transition rates Hij(t) as functions of t. Use your function to calculate numerical values for: (a) A(30) (b) A(50) (c) A(65) [ Over a very short time period h, the transition probability matrix P(t,t+n) is given by the equation: Plt;t+h)=1+hA(t)+o(h) where: . I is the identity matrix, and o(h) is a polynomial in h2 and higher powers of h. Use this equation to code the approximate transition probability matrix P(t,t+h) over a short time period h, and calculate the value of P(34,34.25). Explain how the transition probability matrix, P(30,65), can be calculated using the approximate transition probability matrix P(t,t+h) for different values of t and h. (iv) Use this method to calculate the values of P(30,65) for h=0.01, 0.001 and 0.0001 and comment on the level of accuracy achieved here. (v) Calculate the probability that: (a) (b) (c) a healthy 45-year-old will be sick after 16 years a sick 51-year-old will be healthy after 12 years a sick 20-year-old will be dead after 10 years. ITotal An insurance company is using a Healthy-Sick-Dead (HSD) model, which is a Markov jump proce to monitor the current state of its Income Protection policyholders, with the following transition rate diagram of annual rates as functions of age t : 0.001 + 0.0003 t H: Healthy S: Sick 0.5 -0.005 t 0.0004 + 0.0002 t 0.0009 + 0.0007t D: Dead (i) Code a function to create a generator matrix A(t) that includes the transition rates Hij(t) as functions of t. Use your function to calculate numerical values for: (a) A(30) (b) A(50) (c) A(65) [ Over a very short time period h, the transition probability matrix P(t,t+n) is given by the equation: Plt;t+h)=1+hA(t)+o(h) where: . I is the identity matrix, and o(h) is a polynomial in h2 and higher powers of h. Use this equation to code the approximate transition probability matrix P(t,t+h) over a short time period h, and calculate the value of P(34,34.25). Explain how the transition probability matrix, P(30,65), can be calculated using the approximate transition probability matrix P(t,t+h) for different values of t and h. (iv) Use this method to calculate the values of P(30,65) for h=0.01, 0.001 and 0.0001 and comment on the level of accuracy achieved here. (v) Calculate the probability that: (a) (b) (c) a healthy 45-year-old will be sick after 16 years a sick 51-year-old will be healthy after 12 years a sick 20-year-old will be dead after 10 years. ITotal