13_AS_5B_lec_annotated

# Write down the dierential equation that 1 can be used

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Unformatted text preview: ) x dt dt →0+ µ1j+t x 11 −t px j =1 11 integrating and noting the initial condition 0 px = 1: 11 t px 38/60 = e− t 0 j =1 µ1j+s ds x Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Exercise (continued) (b) Consider j = 2, 3, 4. Write down the diﬀerential equation that 1 can be used to solve for t px j , and hence derive the solution for 1j t px . You may assume that the transition intensities are constant from age x to x + 1. Solution. Assume constant force of mortality over from age x to x + 1. Kolmogorov equation for this model, for 0 &lt; t ≤ 1: ∂ 1j 11 1j t px =t px µx +t ∂t integrating gives (assuming constant piecewise intensity) 1j t px 39/60 = µ1j x 1k k =2,3,4 µx 1 − e− k =1 µ1k t x for 0 ≤ t ≤ 1. Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Exercise (continued) 12 (c) Calculate 2 p40 , given that µ12 1 x+ µ13 1 x+ µ14 1 x+ 40 41 0.2 0.25 0.07 0.04 0.01 0.008 12 2 p40 12 11 =1 p40 +1 p40 x 4 4 4 Solution We have 12 1 p41 and from (b) and using appropriate intensities (µij = µij 1 , µij = µij 1 ) we have 41 40 41+ 40+ 4 4 12 1 p40 11 1 p40 12 1 p41 = 0.174 = 0.756...
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