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# Week5 - 1 ACTL2003 Stochastic Models for Actuarial...

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ACTL2003 Stochastic Models for Actuarial Applications Week 5 Continuous Time Markov Chains- Actuarial Applications Outline: Birth and Death Processes Non-homogeneous Markov jump processes Actuarial Applications - Mortality and Sickness models Readings: Ross, 9th Edition, Chapter 6 (6.1-6.5) Reference: ACTED Chapter 4 and 6 CT4

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Birth and Death Processes A birth and death process is a continuous-time Markov chain with states { 0 , 1 , 2 , . . . } for which transitions from state n may go only to either state n + 1 (a birth) or state n 1 (a death). Suppose that the number of people in a population (or any system) is n and new arrivals enter the population/system at rate λ n (the arrival or birth rate) with the time until the next arrival exponentially distributed with mean 1 λ n , and people leave the population/system at rate μ n (the departure or death rate) with the time until the next departure exponentially distributed with mean 1 μ n and independent of the next arrival.
Birth and Death Processes We have υ 0 = λ 0 υ i = λ i + μ i i > 0 . For the embedded discrete time Markov chain the transition probabilities P ij between the states will be, for state 0 , P 01 = 1 since the number in the system can not be negative and so a birth must be the fi rst jump out of state 0 .

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Birth and Death Processes For i > 0 then the jump will be from i to i + 1 if a birth occurs before a death and from i to i 1 if a death occurs before a birth. The time to a birth, T b , is exponential with rate λ i and the time to a death, T d , is exponential with rate μ i . We require Pr [ T b < T d ] = λ i λ i + μ i . We then have P i,i +1 = λ i λ i + μ i and P i,i 1 = 1 P i,i +1 = μ i λ i + μ i .
Birth and Death Processes Recall from the results for the exponential distribution that if ( X 1 , X 2 , . . . , X n ) are n independent exponential random variables, X i , each with di ff erent rate λ i then the probability that X i is the smallest is λ i P n i =1 λ i . We could have just used this result to derive the transition probabilities above. Recall also that the time to the next jump, regardless of whether it is a birth or a death, is exponential with rate λ i + μ i . Therefore, ν i = λ i + μ i .

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Examples The following are examples of birth and death processes: Birth and death rates independent of n and an example of a queue. A bank (supermarket) has one server and customers join a queue when they arrive. Customers arrive at rate λ with inter-arrival times exponential (a Poisson process) and the server serves customers at the rate μ with service times exponential. If X ( t ) is the number in the queue at time t , then { X ( t ) , t > 0 } is a birth and death process with λ n = λ for n 0 and μ n = μ for n 1 .
Examples What if there are now s tellers (checkout operators) serving the queue in the bank (supermarket) with customers joining a single queue and going to the fi rst available server? We have a birth and death process with λ n = λ for n 0 and μ n = ( nμ, for 1 n < s sμ, for n > s.

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Examples Population growth. Each individual in a population gives birth at an exponential rate λ
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Week5 - 1 ACTL2003 Stochastic Models for Actuarial...

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