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Notations / Definitions:
𝑖𝑗
β€’ 𝑑 𝑝π‘₯ : the probability that someone in state i at time x is in state j (may = i) at time (x+t)
...

𝑖𝑗
β€’ 𝑑𝑝π‘₯𝑖𝑖 ≀ 𝑑 𝑝π‘₯
Μ…Μ…Μ…Μ…
β€’ 𝑑𝑝π‘₯ = 𝑑𝑝π‘₯00
= 𝑑 𝑝π‘₯00
β€’ π‘‘π‘žπ‘₯ = 𝑑 𝑝π‘₯01 (since it is impossible to reenter state 0)
β€’ Markov Chain is a multiple-state model with the following property: the probability of leaving a
state is not a function of the amount of time in the state
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A
function of h, f(h), is o(h) if when it is divided by h, it goes to 0 as h goes to 0
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Discrete Markov chains
1
...
In the matrix, entry ij is the probability of transferring from state i to state j
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2
...
There will be a matrix for every duration
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For an alive person, the probability of transition to state 1 is
π‘ž35+𝑑 , and the probability of transition to state 0 (remaining alive) is 𝑝35+𝑑
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𝑃(𝑑) = ( 35+𝑑
)
0
1
3
...
On the first row, the first entry will be the
probability of survival, the second entry the probability of death from accidental causes, and the third
entry the probability of death from other causes
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01
02
𝑝35+𝑑 𝑝35+𝑑
𝑝35+𝑑
(𝑑)
a
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Consider the disability income model
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𝑃(𝑑) = (𝑝35+𝑑
𝑝35+𝑑
𝑝35+𝑑
0
0
1
5
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Markov Chains Continuous Probability
𝑖𝑗
1
...

𝑖𝑗

𝑝

𝑖𝑗

2
...

4
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β„Žπ‘π‘₯𝑖𝑖 = 1 βˆ’ β„Ž βˆ‘π‘—β‰ π‘– πœ‡π‘₯ + π‘œ(β„Ž)
6
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𝑑𝑝π‘₯𝑖𝑖 = 𝑒 βˆ’ ∫0 βˆ‘π‘—β‰ π‘— πœ‡π‘₯+𝑠𝑑𝑠
b
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Kolmogorov’s forward equations – (i may = j)
𝑛
𝑑 𝑖𝑗
𝑖𝑗 π‘—π‘˜
π‘–π‘˜ π‘˜π‘—
𝑑 𝑝π‘₯ = βˆ‘( 𝑑 𝑝π‘₯ πœ‡π‘₯+𝑑 βˆ’ 𝑑 𝑝π‘₯ πœ‡π‘₯+𝑑 )
𝑑𝑑
π‘˜=0
π‘˜β‰ π‘—

𝑖𝑗

𝑖𝑗

Then pick a small step h and approximate the derivative with (𝑑+β„Ž 𝑝π‘₯ βˆ’ 𝑑 𝑝π‘₯ )/β„Ž, multiplying both
𝑖𝑗
𝑖𝑗
π‘˜π‘—
𝑖𝑗 π‘—π‘˜
side by h, (𝑑+β„Ž 𝑝π‘₯ β‰ˆ 𝑑 π

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