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Unformatted text preview: Review Notes ACTSC 331, FALL 2011 Part 2 Multiple State Models 1. Description of a Multiple State Model (MSM) : Assume that, at any time t , a life aged x has n + 1 states such as alive, dead, employed, unemployed, healthy, sick, disabled, and so on. The n + 1 states are labelled 0 , 1 ,...,n . For any time t , let Y ( t ) denote the states of ( x ) at time t , namely event ( Y ( t ) = i ) means the life is in state i at age x + t . Thus, for a given t 0, Y ( t ) is a discrete random variable with n + 1 possible values. In addition, the set of the random variables { Y ( t ) } t is a continuoustime stochastic process. 2. Assumptions and notation on a multiple state model. Assumption 1 (Markov property): For any states i and j and any t 0 and s 0, the conditional probability Pr { Y ( t + s ) = j  Y ( t ) = i } depends only on any information at the current time t and does not depend on any information before time t . Assumption 2: For any t 0 and h > 0, Pr { 2 or more transitions in the time period ( t, t + h ] } = o ( h ) . A function g ( h ) = o ( h ) means lim h g ( h ) h = 0 . Thus, if g i ( h ) = o ( h ) for i = 1 ,...,k , then k i =1 g i ( h ) = o ( h ), Q k i =1 g i ( h ) = o ( h ), and g i ( h ) = o ( h ), where is an arbitrary constant. Notation: For any states i and j and any x 0 and t 0, we define t p ij x = Pr { Y ( x + t ) = j  Y ( x ) = i } , t p ii x = Pr { Y ( x + s ) = i for all s [0 ,t ]  Y ( x ) = i } ....
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 Fall '09
 david

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