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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B irthandDeathProcesses A birth and death process is a continuous-time Markov chain with states { 0 , 1 , 2 ,... } for which transitions from state n maygoon lytoe ithe rstate 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.
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B irthandDeathProcesses 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 rst jump out of state 0 .
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B irthandDeathProcesses 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 .W erequ i re 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 .
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B irthandDeathProcesses 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 f 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 andanexamp leo faqueue . Abank (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 n 1 .
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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 rst available server? We have a birth and death process with λ n = λ for n 0 and μ n = ( nμ, 1 n<s sμ, n>s.
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Examples Population growth. Each individual in a population gives birth at an exponential rate λ plus there is immigration at an exponential rate θ . Each individual has an exponential
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Week5 - 1 ACTL2003 Stochastic Models for Actuarial...

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