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# 13_AS_5B_lec_annotated - Actuarial Statistics Module 5...

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Actuarial Statistics – Module 5: Parametric methods: Markov Model Actuarial Statistics Benjamin Avanzi c University of New South Wales (2012) School of Risk and Actuarial Studies [email protected] Module 5: Parametric methods: Markov Model 1/60

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Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model General Markov Model (multiple state model) J = { 1 , 2 , . . . , n } finite set of states (state space) S ( t ) continuous time Markov process on states g and h any two states with transition intensity μ gh x + t from state g to state h at age x + t transition probabilities t p gh x = Pr (in state h at age x + t | in state g at age x ) t p gg x = Pr (in state g from age x to age x + t | in state g at age x ) note that in general t p gg x 6 = t p gg x 24/60

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Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Example: Consider a Healthy-Sick-Dead model. Give an example of states such that 1 t p gg x 6 = t p gg x 2 t p gg x = t p gg x 25/60
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Assumptions: Assumptions Markov assumption: the probabilities that the process at any given time will be found in each state at any future time depend only on the times involved and on the state currently occupied For any 2 distinct states, i.e. any g 6 = h dt p gh x + t = μ gh x + t dt + o ( dt ) t 0 Probability of more than one transition in time dt is o ( dt ) 26/60

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Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Kolmogorov Forward Equations: t t p gh x = X j 6 = h h t p gj x μ jh x + t - t p gh x μ hj x + t i Proof For g 6 = h by Markov assumption we have t + dt p gh x = X j 6 = h t p gj x dt p jh x + t + t p gh x dt p hh x + t = X j 6 = h t p gj x dt p jh x + t + t p gh x 1 - X j 6 = h dt p hj x + t = X j 6 = h t p gj x n μ jh x + t dt + o ( dt ) o + t p gh x 1 - X j 6 = h n μ hj x + t dt + o ( dt ) o 27/60
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Proof (continued) We then have t t p gh x = lim dt 0 + t + dt p gh x - t p gh x dt = lim dt 0 + n X j 6 = h t p gj x n μ jh x + t dt + o ( dt ) o + t p gh x 1 - X j 6 = h n μ hj x + t dt + o ( dt ) o - t p gh x o / dt = X j 6 = h t p gj x μ jh x + t - t p gh x X j 6 = h μ hj x + t = X j 6 = h h t p gj x μ jh x + t - t p gh x μ hj x + t i 28/60

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Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Given the transition intensities (estimated from data) we can determine transition probabilities from these equations (Kolmogorov’s Forward Equations) 29/60
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Example The following Markov model has been proposed for a study of unemployment benefits (essentially an annuity payable while a young adult is unemployed for whatever 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

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