{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

13_AS_5B_lec_annotated - Actuarial Statistics Module 5...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Actuarial Statistics – Module 5: Parametric methods: Markov Model Actuarial Statistics Benjamin Avanzi c University of New South Wales (2012) School of Risk and Actuarial Studies [email protected] Module 5: Parametric methods: Markov Model 1/60
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model General Markov Model (multiple state model) J = { 1 , 2 , . . . , n } finite set of states (state space) S ( t ) continuous time Markov process on states g and h any two states with transition intensity μ gh x + t from state g to state h at age x + t transition probabilities t p gh x = Pr (in state h at age x + t | in state g at age x ) t p gg x = Pr (in state g from age x to age x + t | in state g at age x ) note that in general t p gg x 6 = t p gg x 24/60
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Example: Consider a Healthy-Sick-Dead model. Give an example of states such that 1 t p gg x 6 = t p gg x 2 t p gg x = t p gg x 25/60
Background image of page 4
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Assumptions: Assumptions Markov assumption: the probabilities that the process at any given time will be found in each state at any future time depend only on the times involved and on the state currently occupied For any 2 distinct states, i.e. any g 6 = h dt p gh x + t = μ gh x + t dt + o ( dt ) t 0 Probability of more than one transition in time dt is o ( dt ) 26/60
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Kolmogorov Forward Equations: t t p gh x = X j 6 = h h t p gj x μ jh x + t - t p gh x μ hj x + t i Proof For g 6 = h by Markov assumption we have t + dt p gh x = X j 6 = h t p gj x dt p jh x + t + t p gh x dt p hh x + t = X j 6 = h t p gj x dt p jh x + t + t p gh x 1 - X j 6 = h dt p hj x + t = X j 6 = h t p gj x n μ jh x + t dt + o ( dt ) o + t p gh x 1 - X j 6 = h n μ hj x + t dt + o ( dt ) o 27/60
Background image of page 6
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Proof (continued) We then have t t p gh x = lim dt 0 + t + dt p gh x - t p gh x dt = lim dt 0 + n X j 6 = h t p gj x n μ jh x + t dt + o ( dt ) o + t p gh x 1 - X j 6 = h n μ hj x + t dt + o ( dt ) o - t p gh x o / dt = X j 6 = h t p gj x μ jh x + t - t p gh x X j 6 = h μ hj x + t = X j 6 = h h t p gj x μ jh x + t - t p gh x μ hj x + t i 28/60
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Given the transition intensities (estimated from data) we can determine transition probabilities from these equations (Kolmogorov’s Forward Equations) 29/60
Background image of page 8
Actuarial Statistics – Module 5: Parametric methods: Markov Model General Markov Model Example The following Markov model has been proposed for a study of unemployment benefits (essentially an annuity payable while a young adult is unemployed for whatever reason). This model takes the form 3: Death % - 1: Employed 2: Unemployed The transition intensities to the death state are assumed to be the same from both states 1 and 2 and are equal to μ . The transition
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}