Question

QUESTION 4: An insurance company offers a health plan to guard against the financial consequences of a person becoming permanently disabled. For the purpose of this plan the company classifies the health status of its members as being either healthy (state 1), ill (state 2), permanently disabled state 3) or dead (state 4). Only those members who are in either state 1 or state 2 pay into this health plan while only those who are in state 3 receive payments from the plan. The evolution, in time, of the health status of the members of this plan is to be modelled as Markov jump process , {X}}t>0, whose generator matrix, G, is given by: G= [ -0.06 0.36 0 0 0.04 -0.44 0 0 0.01 0.01 0.06 0.02 -0.17 0.17 0 0 where the states have the natural ordering 1, 2, 3, 4 and the unit of time is one year. Do the following: (a) Draw the transition graph of this Markov jump process. [3 marks] (b) For those members who currently are in either state 1 or state 2 calculate the average length of time for which each will be a contributor to this plan. Hint: You may use the fact that if T denotes the first time at which a time-homogeneous Markov jump process, {X2}120, enters some subset A of the state space and if m; = E[T|Xo = i], then the numbers m; obey the equations. S U im; = -1 if ; = 0 if & A i E A. m where G = (9) denotes the generator matrix of this process.

Answer 1

0 600 0이구 2 → 006 3. 0 36 0.0 | --GMD는 Here A = {3,4} - 0.06 m, + 0.0 mg = ____0.36m, - 0. GAM2 | E- + - 0 - 12_m_t_0.16 m, = 0 뿌 3) M = OLE M 아라 구 - 0.06. 0, + 0.0 m, - - ) 6. K of me +0.4mg - -11 _Moo04 ___010.6.4] = -1 | ___m, No.61 2 - AL___ 5 크 깐트3도 m = 40

____ + - 6 ___५५ 36_ ५५ | _- 365 65 +14 ५५ 5 yuX 22 2 5 I Average time will ___M) = 4by Rcts be equal to