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

Bonus-Malus Scales 175
as p n
l 1 l 2
, is the probability of moving from level l 1 to level l 2 in n transitions. Using
Property 4.1, we get the following representation for the distribution of the state variable L n :
denoting as
we have
p k = ( PrL k = 0PrL k = s ) T
p k+n = p k P n (4.6)
Remark 4.1 (Numerical Aspects) The computation of the probability distribution of L n
amounts to calculating the nth power of the transition matrix P. For large values of n, this
may pose some computational difficulties. This is why we now discuss an algebraic method
which makes use of the concept of eigenvalues and eigenvectors (that will be encountered
further in this chapter).
The vector v with at least one component different from 0 is a right eigenvector of P if
Pv = v for some ∈ . In this case, is said to be an eigenvalue of P. Finding the
eigenvalues of P amounts to solving the characteristic equation detP − I = 0. A
nonzero vector u that is a solution of u T P = u T is called a left eigenvector corresponding
to .
In general, the characteristic equation possesses s + 1 solutions 0 s which can be
complex and some of them can coincide (we assume that the eigenvalues are numbered so
that 0 ≥ 1 ≥···≥ s ). The Perron-Froebenius theorem for regular matrices ensures
that
∑
provided the transition matrix P is regular then 0 = 1, v 0 = e, u 0 ≥ 0 with u T e =
s
j=0 u 0j = 1. Moreover, all the other eigenvalues of P lie inside the unit circle of the
complex plane, that is j < 1 for j = 1s.
Let V = v 0 v s be an s +1×s +1 matrix consisting of right column eigenvectors
and
⎛
⎜
U = ⎝
an s + 1 × s + 1 matrix consisting of left column eigenvectors. Let us assume that the
eigenvalues 0 s are distinct. This ensures that v 0 v s are linearly independent so
that V is invertible. Moreover, U = V −1 . In this case, P can be represented as
⎛
⎞
0 ··· 0
⎜
P = V
⎝
⎟
s∑
⎠ U = j v j u T j
j=0
0 ··· s
u T 0
u T s+1
This representation is useful for computing the nth power of P in that
⎛
⎞
n 0
··· 0
P n ⎜
= V ⎝
⎟
s∑
⎠ U = n j v ju T j
0 ··· n j=0
s
⎞
⎟
⎠