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Chapman-Kolmogorov Equations

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Submitted By nrhellwig
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[pic] [pic]

Markov Chain

[pic]

Bonus Malus Model [pic] [pic] This table justifies the matrix above:
| | | |Next state | | |
|State |Premium |0 Claims |1 Claim |2 Claims |[pic]Claims |
|1 | |1 |2 |3 |4 |
|2 | |1 |3 |4 |4 |
|3 | |2 |4 |4 |4 |
|4 | |3 |4 |4 |4 |
| | | | | | |
|P11 |P12 |P13 |P14 | | |
|P21 |P22 |P23 |P24 | | |
|P31 |P32 |P33 |P34 | | |
|P41 |P42 |P43 |P44 | | |
| | | | | | |
|[pic] |[pic] |k |[pic] | | |
| | | | | | |
|P11 |S1(k) =1 |0 |[pic] | | |
|P12 |S1(k) =2 |1 |[pic] | | |
|P13 |S1(k) =3 |2 |[pic] | | |
|P14 |S1(k) =4 |3, 4,… |[pic] |[pic] | |
|P21 |S2(k) =1 |0 |[pic] | | |
|P22 |S2(k) =2 | |0 | | |
|P23 |S2(k) =3 |1 |[pic] | | |
|P24 |S2(k) =4 |2, 3, 4,… |[pic]+… |[pic] | |
|P31 |S3(k) =1 | |0 | | |
|P32 |S3(k) =2 |0 |[pic] | | |
|P33 |S3(k) =3 | |0 | | |
|P34 |S3(k) =4 |1,2,3,4, … |[pic]+… |[pic] | |
|P41 |S4(k) =1 | |0 | | |
|P42 |S4(k) =2 | |0 | | |
|P43 |S4(k) =3 |0 |[pic] | | |
|P44 |S4(k) =4 |1,2,3,4, … |[pic]+… |[pic] | |

Chapman-Kolmogorov Equations We have already defined the one-step transition probabilities [pic]. We now define the n-step transition probabilities [pic] to be the probability that a process in state i will be in state j after n additional transitions. That Is, [pic][pic] n, i,j[pic] 0. Of course [pic]=[pic]. The Chapman-Kolmogorov equations provide a method for computing these n-step transition probabilities. These equations are [pic]

and are most easily understood by noting that [pic] represents the probability that starting in i the process will go to state j in n -1- m transitions through a path which takes it into state k at the nth transition. Hence, summing over all intermediate states k yields the probability that the process will be in state j after n+m transitions. Formally, we have [pic]

Remarks: 1. To obtain the 4th equality, we have used the multiplication rule of conditional probabilities and the result that [pic] = [pic]: [pic] ([pic]=[pic]=1) 2. [pic] is not a matrix, It is the entry in the i row and the k column of the matrix [pic].

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