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4 dead the transition intensities from state i to

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Unformatted text preview: Markov Model General Markov Model Example: Exam Style Question Consider the following Markov model (and the respective intensities) used for the modelling of a TPD (Total and Permanent Disability) insurance 1: Able ↓ 3: Lapsed 2: Permanently Disabled. 4: Dead The transition intensities from state i to state j at age x + s are represented by µij +s . x 11 (a) Derive from ﬁrst principles a diﬀerential equation for t px , and show that the solution is t µ12 s + µ13 s + µ14 s ds x+ x+ x+ 11 t px = exp − 0 36/60 You should state any assumptions that you make. Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Exercise (continued) (a) Solution Assumptions Markov assumption - current state (and age) is all that is needed to determine probability of future state For any 2 distinct states gh dt px +t = µgh t dt + o (dt ) x+ t≥0 Probability of more than one transition in time dt is o (dt ) Hence 11 t +dt px = 11 t px 1 1j dt px +t − j =1 = 37/60 11 t px 1 µ1j+t dt + o (dt ) x − j =1 Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Exercise (continued) rearranging and taking limits ∂ 11 tp ∂t x = = = lim + 11 t +dt px 11 −t px dt dt →0 11 −t px lim j =1 µ1j+t dt + o (dt...
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