Lecture4 SIQ3003.pdf
- 1. Stochastic Process and Markov Chain Discrete Markov Chain Actuarial Mathematics II Lecture 4 Dr. Shaiful Anuar Institute Of Mathematical Sciences University of Malaya Lecture 4 Actuarial Mathematics II
- 2. Stochastic Process and Markov Chain Discrete Markov Chain Table of contents 1 Stochastic Process and Markov Chain 2 Discrete Markov Chain Example Example Example Example Lecture 4 Actuarial Mathematics II
- 3. Stochastic Process and Markov Chain Discrete Markov Chain Stochastic Process and Markov Chain A collection of random variables of a stochastic process is normally represented by {Yt, t ∈ T} where t is often referred to as time and Yt is the process state at time t with Yt ∈ S. S is known as the state space, set of all possible values of the random variables. The stochastic process can be categorized as follows depending on T: discrete-time stochastic process : when T is of discrete/countable type continuous-time stochastic process : when T is of continuous/interval type Similarly a Markov chain can be of discrete or continuous type. Lecture 4 Actuarial Mathematics II
- 4. Stochastic Process and Markov Chain Discrete Markov Chain Markov chain is a stochastic process with the following properties: (i) P(Yt+1 ∈ S|Yt = i) = 1 (ii) P(Yt+1 = j|Yn = i, Yn−1 = in−1, · · · , Y0 = i0) = Pr(Yt+1 = j|Yt = i) An important characteristics of a Markov chain as expressed in property (ii) above is the memoryless property, that is, the transition probabilities only depend on the current state. In insurance, we apply this concept in multi-state models which uses the concept of Markov chain to specify the probabilities of moving to various states in the model. The probabilities are known as transition probabilities. Lecture 4 Actuarial Mathematics II
- 5. Stochastic Process and Markov Chain Discrete Markov Chain Several popular multi-state models are given below: (i) The alive-dead model Alive (1) Dead (2) (ii) The double indemnity/accidental death model Alive (1) Dead - Accident (2) Dead - Other causes (3) Lecture 4 Actuarial Mathematics II
- 6. Stochastic Process and Markov Chain Discrete Markov Chain (iii) The permanent disability model Healthy (1) Disabled (2) Dead (3) (iv) The disability income model Healthy (1) Sick (2) Dead (3) Lecture 4 Actuarial Mathematics II
- 7. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Discrete Markov Chain The most common model that we have come across is the single decrement model with a two-state transition. The alive-dead model Alive (1) Dead (2) It is common to represent the probability of transition in the form of a matrix. For the above example, assume that the probability of moving from state ”Alive” and ”Death” equals 0.4. Then, the matrix can be represented as follows: 1 2 1 0.6 0.4 2 0 1 Lecture 4 Actuarial Mathematics II
- 8. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example One-step transition probability is the probability of moving from one state to another in a single step. It is denoted by: Q (i,j) t = P(Yt+1 = j|Yt = i) kth-step transition probability is the probability of moving from one state to another in a single step. It is denoted by: kQ (i,j) t = X k Q (i,k) t · Q (k,j) t+1 As mentioned earlier, the transition probabilities is given in a matrix known as transition matrix. If the transition probabilities does not vary with time, it is referred to as homogeneous Markov chain. If the transition probabilities varies with time, it is referred to as non-homogeneous Markov chain. Lecture 4 Actuarial Mathematics II
- 9. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example For a homogeneous Markov chain the transition probability is independent of t. kQ (i,j) t = X k Q(i,k) · Q(k,j) Therefore if Q is the transition matrix, then the kth-step transition matrix is simply given by: kQt = Qk Lecture 4 Actuarial Mathematics II
- 10. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Example 1 For a homogeneous Markov chain, the transition probability matrix is given by 1 2 1 0.4 0.6 2 0.8 0.2 Calculate 3Q(2,1). Lecture 4 Actuarial Mathematics II
- 11. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Solution Lecture 4 Actuarial Mathematics II
- 12. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Example 2 An auto insured who was claim-free during a policy period will claim-free during the next policy period with probability 0.9. An auto insured who was not claim-free during a policy period will be claim-free during the next policy period with probability 0.7. What is the probability that an insured who was claim-free during the policy period 0 will incur a claim during period 3? Lecture 4 Actuarial Mathematics II
- 13. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Solution Lecture 4 Actuarial Mathematics II
- 14. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Example 3 Consider a homogeneous Markov model with three states, Healthy (0), Disabled (1) and Dead (2). (i) The annual transition matrix is given by 0 1 2 0 0.7 0.2 0.10 1 0.10 0.65 0.25 2 0 0 1 (ii) There are 100 lives at the start, all Healthy. Their future states are independent. (iii) Assume that all lives have the same age at start. Calculate the variance of the number of the original lives who die within the first two years. Lecture 4 Actuarial Mathematics II
- 15. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Solution Lecture 4 Actuarial Mathematics II
- 16. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Example 4 A life insurance policy waives premium upon disability of the insured. The policy is modeled as a homogeneous Markov chain with the three states: active, disabled, gone. The annual transition matrix is 1 2 3 1 0.75 0.15 0.10 2 0.50 0.30 0.20 3 0 0 1 Currently 90% of the insureds are active and 10% are disabled. (a) Calculate the percentage of the current population of insureds that are (1) active, (2) disabled and (3) gone at the end of three years. (b) Calculate the probability that a currently disabled life will be active at the end of three years. Lecture 4 Actuarial Mathematics II
- 17. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Solution Lecture 4 Actuarial Mathematics II
- 18. Stochastic Process and Markov Chain Discrete Markov Chain Example Example Example Example Solution Lecture 4 Actuarial Mathematics II