Dene wi the duration until the process leaves state i

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Unformatted text preview: = 0.216 so 40/60 12 2 p40 = 0.338 Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Holding times and jump probabilities- a review In a time-homogeneous Markov jump process, let µij be the transition rates from state i to state j for j = i and write µi = j =i µij . define Wi : the duration until the process leaves state i given that the current state is i . Then: Wi is an exponential random variable, SWi (t ) = e −µi t FWi (t ) = 1 − e −µi t fWi (t ) = µi e −µi t 1 E (Wi ) = µi Var (Wi ) = 41/60 1 µi 2 Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model The probabilities that the Makov process {Xt : t ≥ 0} goes into state j when it leaves state i : P (XWi = j |X0 = i ) = where µi is defined by µi = 42/60 j =i µij . µij , µi Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Markov process-review Example Assume that i can only go to i + 1 and µi is the transition intensity from i to i + 1. Probability that a person in state i will be in state i + 1 in t years from now t µi e −µi s e −µi +1 (t −s ) ds 0 = µi e −µi +1 t e −µi t + µi +1 − µi µi − µi +1 Note - probability jump out of state i between time s and s + ds is µi e −µi s ds and probability stay in state i + 1 from s to t is e −µi +1 (t −s ) 43/60 Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov M...
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This document was uploaded on 04/03/2014.

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