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Unformatted text preview: to estimate
the transition probabilities of a Total and Permanent Disability
Insurance Contract: 2: Permanently Disabled. → 1: Able
3: Dead The (constant) transition intensity from state i to state j is
denoted as µij and the corresponding transition probability (for
time period t ) is denoted by Pij (t ).
(a) Write down the Kolmogorov equations that can be used to ﬁnd
the transition probabilities in this model.
(b) Hence or otherwise derive in closed form all the transition
probabilities in this model.
32/60 Actuarial Statistics – Module 5: Parametric methods: Markov Model
General Markov Model Solution (a) Kolmogorov’s equations for the non-trivial transitions:
P11 (t ) = −P11 (t ) (µ12 + µ13 ) (1) P12 (t ) = P11 (t ) µ12 − P12 (t ) µ23 (2) P13 (t ) = P11 (t ) µ13 + P12 (t ) µ23 (3) P22 (t ) = −P22 (t ) µ23 (4) P23 (t ) = P22 (t ) µ23 (5) (b) Notice that It is not necessary to solve all 5 equations directly.
Solving (1) gives P11 (t ) = e −(µ12 +µ13 )t .
Solving (5) gives P22 (t ) = e −µ23 t
and as P21 (t ) = 0, we have P23 (t ) = 1 − e −µ23 t .
Now we solve (2). First re-arrange and substitute P11 (t ) to get
P12 (t ) = −µ23 P12 (t ) + µ12 e −(µ12 +µ13 )t
33/60 Actuarial Statistics – Module 5: Parametric methods: Markov Mo...
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