This model takes the form 3 death 1 employed 2

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Unformatted text preview: atever reason). This model takes the form 3: Death 1: Employed 2: Unemployed The transition intensities to the death state are assumed to be the same from both states 1 and 2 and are equal to µ. The transition probability from state 1 to state 2 for this model is given by P12 (t ) = µ12 e −µt + e −(µ+µ12 +µ21 )t µ12 + µ21 Derive an expression for P11 (t ) in terms of a function involving the transition intensities and the time t . 30/60 Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Example (continued) Solution. From Kolmogorov equation, P12 (t ) = P11 (t ) µ12 + P12 (t ) (−µ − µ21 ) hence P11 (t ) = P12 (t ) + P12 (t ) (µ + µ21 ) µ12 but we have P12 (t ) = µ12 e −µt + e −(µ+µ12 +µ21 )t µ12 + µ21 hence P12 (t ) = µ12 −µe −µt − (µ + µ12 + µ21 ) e −(µ+µ12 +µ21 )t µ12 + µ21 on substitution P11 (t ) = 31/60 1 µ21 e −µt + µ12 e −(µ+µ12 +µ21 )t µ12 + µ21 Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Exercise The following Markov model is to be used...
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