In this section we focus on efficient simulation of the claim arrival point process . This process can be simulated either via the arrival times
, i.e. moments when the
th claim occurs, or the inter-arrival times (or waiting times)
, i.e. the time periods
between successive claims. Note that in terms of
's the claim arrival point process is given by
. In what
follows we discuss five prominent examples of
, namely the classical (homogeneous) Poisson process, the non-homogeneous Poisson process,
the mixed Poisson process, the Cox process (also called the doubly stochastic Poisson process) and the renewal process.
The most common and best known claim arrival point process is the homogeneous Poisson process (HPP) with stationary and independent increments and the number of claims in a given time interval governed by the Poisson law. While this process is normally appropriate in connection with life insurance modeling, it often suffers from the disadvantage of providing an inadequate fit to insurance data in other coverages. In particular, it tends to understate the true variability inherent in these situations.
Formally, a continuous-time stochastic process
is a (homogeneous) Poisson process with intensity (or rate)
if (i)
is a point process, and (ii) the waiting times
are independent and identically distributed and follow an exponential law with intensity
, i.e. with mean
(see Chapter 13, where the properties and simulation scheme for the exponential distribution were discussed). This definition naturally leads to a simulation scheme for the successive arrival times
of the Poisson process:
Alternatively, the homogeneous Poisson process can be simulated by applying the following property (Rolski et al.; 1999). Given that ,
the
occurrence times
have the same distributions as the order statistics corresponding to
i.i.d. random variables
uniformly distributed on the interval
. Hence, the arrival times of the HPP on the interval
can be generated as follows:
In general, this algorithm will run faster than the previous one as it does not involve a loop. The only two inherent numerical difficulties
involve generating a Poisson random variable and sorting a vector of occurrence times. Whereas the latter problem can be solved via the standard
quicksort algorithm, the former requires more attention. A simple algorithm would take
,
which is a consequence of the properties of the Poisson process (for a derivation see Ross; 2002). However, for large
, this
method can become slow. Faster, but more complicated methods have been proposed in the literature. Ahrens and Dieter (1982) suggested a generator
which utilizes acceptance-complement with truncated normal variates whenever
and reverts to table-aided inversion otherwise.
Stadlober (1989) adapted the ratio of uniforms method for
and classical inversion for small
's. Hörmann (1993) advocated the transformed rejection method, which is a combination of the inversion and rejection algorithms.
Sample trajectories of homogeneous and non-homogeneous Poisson processes are plotted in Figure 14.1. The dotted green line is a HPP with intensity (left panel) and
(right panel). Clearly the latter
jumps more often. Since for the HPP the expected value
, it is natural to define the premium function in this
case as
, where
,
and
is the relative safety loading which
``guarantees'' survival of the insurance company. With such a choice of the premium function we obtain the classical form of the risk process.
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The choice of a homogeneous Poisson process implies that the size of the portfolio cannot increase or decrease. In addition, it cannot describe situations, like in motor insurance, where claim occurrence epochs are likely to depend on the time of the year or of the week. For modeling such phenomena the non-homogeneous Poisson process (NHPP) suits much better than the homogeneous one. The NHPP can be thought of as a Poisson process with a variable intensity defined by the deterministic intensity (rate) function
. Note that the increments of a NHPP do not have to be stationary. In the special case when
takes the constant value
, the NHPP reduces to the homogeneous Poisson process with intensity
.
The simulation of the process in the non-homogeneous case is slightly more complicated than in the homogeneous one. The first approach, known as the
thinning or rejection method, is based on the following fact (Ross; 2002; Bratley, Fox, and Schrage; 1987). Suppose that there exists a constant
such that
for all
. Let
be the successive arrival times of
a homogeneous Poisson process with intensity
. If we accept the
th arrival time
with probability
, independently of all other arrivals, then the sequence
of the accepted arrival times (in ascending order)
forms a sequence of the arrival times of a non-homogeneous Poisson process with the rate function
. The resulting algorithm reads as
follows:
Note that in the above algorithm we generate a HPP with intensity
employing the HPP1 algorithm. We can also generate it using
the HPP2 algorithm, which is in general much faster.
The second approach is based on the observation (Grandell; 1991) that for a NHPP with rate function
the increment
,
,
is distributed as a Poisson random variable with intensity
. Hence, the cumulative distribution
function
of the waiting time
is given by
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The third approach utilizes a generalization of the property used in the HPP2 algorithm. Given that , the
occurrence times
of the non-homogeneous Poisson process have the same distributions as the order statistics corresponding to
independent random variables distributed on the interval
, each with the common density function
, where
. Hence, the arrival times of the NHPP on the interval
can be generated as follows:
The performance of the algorithm is highly dependent on the efficiency of the computer generator of random variables with density . Moreover, like in the homogeneous case, this algorithm has the advantage of not invoking a loop. Hence, it performs faster than the former two methods if
is a nicely integrable function.
Sample trajectories of non-homogeneous Poisson processes are plotted in Figure 14.1. In the left panel realizations of a NHPP with
linear intensity
are presented for the same value of parameter
. Note, that the higher the value of parameter
, the
more pronounced is the increase in the intensity of the process. In the right panel realizations of a NHPP with periodic intensity
are illustrated, again for the same value of parameter
. This time, for high values of parameter
the
events exhibit a seasonal behavior. The process has periods of high activity (grouped around natural values of
) and periods of low activity,
where almost no jumps take place. Finally, we note that since in the non-homogeneous case the expected value
, it is natural to define the premium function as
.
In many situations the portfolio of an insurance company is diversified in the sense that the risks associated with different groups of policy holders are significantly different. For example, in motor insurance we might want to make a difference between male and female drivers or between drivers of different age. We would then assume that the claims come from a heterogeneous group of clients, each one of them generating claims according to a Poisson distribution with the intensity varying from one group to another.
Another practical reason for considering yet another generalization of the classical Poisson process is the following. If we measure the volatility of risk processes, expressed in terms of the index of dispersion
, then very often we obtain estimates in excess of one - a value obtained for the homogeneous and the non-homogeneous cases. These empirical observations led to the introduction of the mixed Poisson process (Ammeter; 1948).
In the mixed Poisson process the distribution of is given by a mixture of Poisson processes (Rolski et al.; 1999). This means that, conditioning on an extrinsic random variable
(called a structure variable), the process
behaves like a homogeneous Poisson process.
Since for each
the claim numbers
up to time
are Poisson variates with intensity
, it is now reasonable to consider the premium function of the form
.
The process can be generated in the following way: first a realization of a non-negative random variable is generated and, conditioned
upon its realization,
as a homogeneous Poisson process with that realization as its intensity is constructed. Both the HPP1 and the HPP2
algorithm can be utilized. Making use of the former we can write:
The Cox process, or doubly stochastic Poisson process, provides flexibility by letting the intensity not only depend on time but also by allowing it to be a stochastic process.
Therefore, the doubly stochastic Poisson process can be viewed as a two-step randomization procedure. An intensity process
is used to generate another process
by acting as its intensity. That is,
is a Poisson process conditional on
which itself is a stochastic process. If
is deterministic, then
is a non-homogeneous Poisson process. If
for some positive random variable
, then
is a mixed Poisson process.
In the doubly stochastic case the premium function is a generalization of the former functions, in line with the generalization of the claim arrival process. Hence, it takes the form
.
The definition of the Cox process suggests that it can be generated in the following way: first a realization of a non-negative stochastic process
is generated and, conditioned upon its realization,
as a non-homogeneous Poisson process with that realization as its
intensity is constructed. Out of the three methods of generating a non-homogeneous Poisson process the NHPP1 algorithm is the most general and,
hence, the most suitable for adaptation. We can write:
Generalizing the homogeneous Poisson process we come to the point where instead of making non-constant, we can make a variety of different
distributional assumptions on the sequence of waiting times
of the claim arrival point process
. In some particular
cases it might be useful to assume that the sequence is generated by a renewal process, i.e. the random variables
are i.i.d. and positive. Note
that the homogeneous Poisson process is a renewal process with exponentially distributed inter-arrival times. This observation lets us write the
following algorithm for the generation of the arrival times of a renewal process:
An important point in the previous generalizations of the Poisson process was the possibility to compensate risk and size fluctuations by the premiums. Thus, the premium rate had to be constantly adapted to the development of the claims. For renewal claim arrival processes, a constant premium rate allows for a constant safety loading (Embrechts and Klüppelberg; 1993). Let be a renewal process and assume that
has finite mean
. Then the premium function is defined in a natural way as
, like for the homogeneous Poisson process.