Address the following questions appropriately. Ensure precision in your answers.

A life insurance company uses the three-state illness-death model as shown below to calculate premiums for a 2-year sickness policy issued to healthy policyholders aged 58. H = healthy S = sick Px D = dead S, denotes the state occupied by the policyholder at age 58+1, so that So = H (healthy) and S, = H, S or D (healthy, sick or dead) for f = 1, 2. The transition intensities used by the insurer are defined in the following way: PSSHI = P(S,+ = KIS, = j). 1 =0,1 For / = 0, 1, it is assumed that: HH P5847 = 0.92 HS P58+ = 0.05 P58+ = 0.65 P58+1 = 0.25 The policy provides a benefit of $5,000 at the end of each year if the policyholder is then sick, and a benefit of $20,000 at the end of the year of death. Calculate the expected present value of the benefits under this policy, assuming an interest rate of 5% pa. [5]{X, } is a Markov jump process with state space $ = (0,1,2,..., and X, =0. The transition rates are given by: A if j=i+1 ME- ifj=i 0 otherwise (i) Write down the transition probabilities P (). (ii) Define the term holding time. [1] (iii) Find the distribution of the first holding time To- (iv) State the value of XT. - [1] (v) Given that the increments are stationary and independent, state the distributions of To. Ti. T2..... Justify your answer. [3] [Total 9]