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

182 Actuarial Modelling of Claim Counts
4.4.3 Dufresne Algorithm
Dufresne (1988,1995) proposed a simple and efficient iterative algorithm for deriving
provided that the driver goes one level down if no claims are filed to the company, and goes
n×pen levels up if n claims are reported to the insurer. Then, the move at the end of year k
can be modelled as
{ −1 if no claims
k+1 =
n × pen if n claims
If the annual numbers of claims N 1 N 2 are independent and oi distributed then the
sequence 1 2 is made up of independent and identically distributed random
variables, with common probability mass function
Pr k+1 =−1 = PrN k+1 = 0 = exp−
Pr k+1 = n × pen = PrN k+1 = n = exp− n
n!
Pr k+1 = = 0 otherwise
The level L k+1 can then be represented as
for n = 1 2
⎧
⎪⎨ L k + k+1 if 0 ≤ L k + k+1 ≤ s
L k+1 = 0ifL k + k+1 =−1
⎪⎩
s if L k + k+1 >s
Let us denote as F k · the distribution function of L k , that is
F k l = PrL k ≤ l l = 0 1s
Furthermore, let us denote as p · the common probability mass function of the k s. We
then have
F k+1 l =
l∑
F k l − yp y
y=−1
with F k+1 s = 1. The stationary distribution F · is then obtained as
F l = lim F kl =
k→+
l∑
F l − yp y
with F s = 1. Obviously, the l s are then recovered from 0 = F 0 and
y=−1
l = F l − F l − 1 for l = 1s
The values of F l can be computed recursively from the following algorithm: