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

Mixed Poisson Models for Claim Numbers 25
Remark 1.1 Note that a better notation would have been oi F instead of
oi since only the distribution function of matters to define the associated
Poisson mixture. We have nevertheless opted for oi for simplicity.
Note that the condition E = 1 ensures that when N ∼ oi
EN =
∫ +∑
0
k=0
= E =
k exp− k dF
k!
or, more briefly,
[ ]
EN = E EN = E = (1.28)
In (1.28), E· means that we take an expected value considering as a constant. We
then average with respect to all the random components, except . Consequently, E· is
a function of . Given , N is Poisson distributed with mean so that EN = . The
mean of N is finally obtained by averaging EN with respect to . The expectation of N
given in (1.28) is thus the same as the expectation of a oi distributed random variable.
Taking the heterogeneity into account by switching from the oi to the oi
distribution has no effect on the expected claim number.
1.4.3 Mixed Poisson Process
The Poisson processes are suitable models for many real counting phenomena but they are
insufficient in some cases because of the deterministic character of their intensity function.
The doubly stochastic Poisson process (or Cox process) is a generalization of the Poisson
process when the rate of occurrence is influenced by an external process such that the
rate becomes a random process. So, the rate, instead of being constant (homogeneous
Poisson process) or a deterministic function of time (nonhomogeneous Poisson process)
becomes itself a stochastic process. The only restriction on the rate process is that it
has to be nonnegative. Mixed Poisson distributions are linked to mixed Poisson processes
in the same way that the Poisson distribution is associated with the standard Poisson
process.
Specifically, let us assume that given = , Nt t ≥ 0 is a homogeneous Poisson
process with rate . Then Nt t ≥ 0 is a mixed Poisson process, and for any s t ≥ 0,
the probability that k events occur during the time interval s t is
PrNt + s − Ns = k =
=
∫
0
∫
0
PrNt + s − Ns = k = dF
exp−t tk dF
k!
that is, Nt + s − Ns ∼ oit . Note that, in contrast to the Poisson process, mixed
Poisson processes have dependent increments. Hence, past number of claims reveal future
number of claims in this setting (in contrast to the Poisson case).