Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

174 Actuarial Modelling of Claim Counts
4.3.3 Multi-Step Transition Probabilities
The probability
p n
ij = PrL k+n = jL k = i
evaluates the likelihood of being transferred from level i to level j in n steps. Note that this
is the probability that L k+n = j given L k = i for any k. The process describing the
trajectory of the policyholder accross the levels is thus stationary. From
p n
ij =
s∑
i 1 =0 i 2 =0
s∑
s∑
i n−1 =0
p ii1 p i1 i 2
···p in−1 j
we clearly see that it includes all the possible paths from i to j and the probability of their
occurrence. This is the n-step transition probability p n
ij . Therefore, the matrix
⎛
P n =
⎜
⎝
p n
00
p n
10
pn 01
pn 11
p n
s0
··· pn 0s
⎞
··· pn 1s
⎟
⎠
pn s1 ··· pn
ss
is called the n-step transition matrix corresponding to P.
The following result shows that P n is a stochastic matrix, being the nth power of the
one-step transition matrix P.
Property 4.1
For all n m = 0 1,
P n = P n (4.3)
and hence,
P n+m = P n P m (4.4)
Proof The proof is by induction on n. The result is obviously true for n = 1. Assume it
holds for n and let us show that it is still true for n+1. Clearly, by conditioning on the level
l occupied at time n we get
p n+1
ij =
s∑
l=0
p n
il p lj (4.5)
which corresponds to matrix multiplication. This proves (4.3), from which (4.4) readily
follows.
□
The matrix identity (4.4) is usually called the Chapman Kolmogorov equation. Taking
the nth power of P yields the n-step transition matrix whose element l 1 l 2 , denoted