The following diagram shows the transitions under a Healthy-Sick-Dead multiple state model under which: transition rates are dependent on time, t. transitions out of the Sick state are dependent on the duration, C, a person has been in the Sick state as well as on time. (f) Healthy Sick p(1. C) (7) v(t, C) Dead (i) Show from first principles, that if p(x,t) is the probability of being in state at time t conditional on being in state at time x, that: PHH(x.1) PHH(x,t)(-(t)-(t))+PHS(x,t)p(t,C) at [5] (ii) Determine the probability that a life is in the Healthy state throughout the period 0 to 1 if the life is in the Healthy state at time 0. [2] (iii) Describe how integrated Kolmogorov equations can be constructed by conditioning on the first or the last jump, illustrating your answer with a diagram. 31 (iv) Explain the difference in approach between deriving forward and backward integrated Kolmogorov equations. [1] An actuarial student suggests the following integrated Kolmogorov equation for this model: = P[X,-HX, S.C,-w] (p(uw-s+u)+D(uw-s+u)\du se (y.w-s+y)PHH(y,t)dy (v) Identify TWO errors in this equation. 12 [Total 13]