E any g h gh dt px t gh t dt o dt x t0 probability

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Unformatted text preview: . any g = h gh dt px +t = µgh t dt + o (dt ) x+ t≥0 Probability of more than one transition in time dt is o (dt ) 26/60 Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Kolmogorov Forward Equations: ∂ gh gj jh gh hj = tp t px µx +t −t px µx +t ∂t x j =h Proof For g = h by Markov assumption we have gh t +dt px = jh gj t px dt px +t gh +t px hh dt px +t j =h jh gj t px dt px +t = hj dt px +t gh +t px 1 − j =h j =h gj t px = µjh+t dt + o (dt ) x j =h 27/60 gh +t px 1 − µhj+t dt + o (dt ) x Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Proof (continued) We then have ∂ gh tp ∂t x = = lim + gh t +dt px gh −t px dt dt →0 gj t px lim + dt →0 µjh+t dt + o (dt ) x j =h gh µhj+t dt + o (dt ) −t px /dt x gh +t px 1 − j =h gj t px = j =h j =h 28/60 µhj+t x j =h gj t px = gh µjh+t −t px x gh µjh+t −t px µhj+t x x 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 wh...
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