13_AS_5B_lec_annotated

# Dene wi the duration until the process leaves state i

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 04/03/2014.

Ask a homework question - tutors are online