14.2 Claim Arrival Processes

In this section we focus on efficient simulation of the claim arrival point process $ \{N_t\}$. This process can be simulated either via the arrival times $ \{T_i\}$, i.e. moments when the $ i$th claim occurs, or the inter-arrival times (or waiting times) $ W_i=T_i-T_{i-1}$, i.e. the time periods between successive claims. Note that in terms of $ W_i$'s the claim arrival point process is given by $ N_t=\sum_{n=1}^\infty I(T_n \le t)$. In what follows we discuss five prominent examples of $ \{N_t\}$, 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.


14.2.1 Homogeneous Poisson 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 $ \{N_t: t \ge 0 \}$ is a (homogeneous) Poisson process with intensity (or rate) $ \lambda > 0$ if (i) $ \{N_t\}$ is a point process, and (ii) the waiting times $ W_i$ are independent and identically distributed and follow an exponential law with intensity $ \lambda $, i.e. with mean $ 1/\lambda$ (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 $ T_1,T_2,\ldots,T_n$ of the Poisson process:

Algorithm HPP1
Step 1:
set $ T_0 = 0$
Step 2:
for $ i = 1,2,\ldots,n$ do
Step 2a:
generate an exponential random variable $ E$ with intensity $ \lambda $
Step 2b:
set $ T_i = T_{i-1} + E$

Alternatively, the homogeneous Poisson process can be simulated by applying the following property (Rolski et al.; 1999). Given that $ N_t=n$, the $ n$ occurrence times $ T_1,T_2,\ldots,T_n$ have the same distributions as the order statistics corresponding to $ n$ i.i.d. random variables uniformly distributed on the interval $ (0,t]$. Hence, the arrival times of the HPP on the interval $ (0,t]$ can be generated as follows:

Algorithm HPP2
Step 1:
generate a Poisson random variable $ N$ with intensity $ \lambda $
Step 2:
generate $ N$ random variables $ U_i$ distributed uniformly on $ (0,1)$, i.e. $ U_i\sim \textrm{U}(0,1),i=1,2,\ldots,N$
Step 3:
set $ (T_1,T_2,\ldots,T_N)=t\cdot\textrm{sort}\{U_1,U_2,\ldots,U_N\}$

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 $ N = \min\{n:U_1\cdot \ldots \cdot U_n < \exp(-\lambda)\}-1$, which is a consequence of the properties of the Poisson process (for a derivation see Ross; 2002). However, for large $ \lambda $, 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 $ \lambda > 10$ and reverts to table-aided inversion otherwise. Stadlober (1989) adapted the ratio of uniforms method for $ \lambda > 5$ and classical inversion for small $ \lambda $'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 $ \lambda =1$ (left panel) and $ \lambda=10$ (right panel). Clearly the latter jumps more often. Since for the HPP the expected value $ \mathop{\textrm{E}} (N_t)=\lambda t$, it is natural to define the premium function in this case as $ c(t)=ct$, where $ c=(1+\theta)\mu\lambda$, $ \mu = \mathop{\textrm{E}} (X_k)$ and $ \theta>0$ 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.

Figure 14.1: Left panel: Sample trajectories of a NHPP with linear intensity $ \lambda (t)=a+b\cdot t$ for $ a=1$ and $ b=1$ (solid blue line), $ b=0.1$ (dashed red line), and $ b=0$ (dotted green line). Note that the latter is in fact a HPP. Right panel: Sample trajectories of a NHPP with periodic intensity $ \lambda(t)=a+b\cdot \cos(2\pi t)$ for $ a=10$ and $ b=10$ (solid blue line), $ b=1$ (dashed red line), and $ b=0$ (dotted green line). Again, the latter is a HPP.
\includegraphics[width=.7\defpicwidth]{STFrisk01a.ps} \includegraphics[width=.7\defpicwidth]{STFrisk01b.ps}


14.2.2 Non-homogeneous Poisson Process

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 $ \lambda(t)$. Note that the increments of a NHPP do not have to be stationary. In the special case when $ \lambda(t)$ takes the constant value $ \lambda $, the NHPP reduces to the homogeneous Poisson process with intensity $ \lambda $.

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 $ \overline{\lambda}$ such that $ \lambda(t)\leq\overline{\lambda}$ for all $ t$. Let $ T_1^* ,T_2^* , T_3^* ,\ldots$ be the successive arrival times of a homogeneous Poisson process with intensity $ \overline{\lambda}$. If we accept the $ i$th arrival time $ T_i^*$ with probability $ \lambda(T_i^*
)/\overline{\lambda}$, independently of all other arrivals, then the sequence $ T_1, T_2, \ldots$ 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 $ \lambda(t)$. The resulting algorithm reads as follows:

Algorithm NHPP1 (Thinning)
Step 1:
set $ T_0 = 0$ and $ T^* = 0$
Step 2:
for $ i = 1,2,\ldots,n$ do
Step 2a:
generate an exponential random variable $ E$ with intensity $ \overline{\lambda}$
Step 2b:
set $ T^* = T^* + E$
Step 2c:
generate a random variable $ U$ distributed uniformly on $ (0,1)$
Step 2d:
if $ U>\lambda(T^*)/\overline{\lambda}$ then return to step 2a ( $ \rightarrow$ reject the arrival time) else set $ T_i = T^*$ ( $ \rightarrow$ accept the arrival time)
As mentioned in the previous section, the inter-arrival times of a homogeneous Poisson process have an exponential distribution. Therefore steps 2a-2b generate the next arrival time of a homogeneous Poisson process with intensity $ \overline{\lambda}$. Steps 2c-2d amount to rejecting (hence the name of the method) or accepting a particular arrival as part of the thinned process (hence the alternative name).

Note that in the above algorithm we generate a HPP with intensity $ \overline{\lambda}$ 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 $ \lambda(t)$ the increment $ N_t-N_s$, $ 0 < s < t$, is distributed as a Poisson random variable with intensity $ \widetilde{\lambda} = \int_s^t \lambda(u)du$. Hence, the cumulative distribution function $ F_s$ of the waiting time $ W_s$ is given by

$\displaystyle F_s (t)$ $\displaystyle =$ $\displaystyle \textrm{P} (W_s \leq t) = 1- \textrm{P}(W_s > t) = 1- \textrm{P}(N_{s+t} - N_s = 0) =$  
  $\displaystyle =$ $\displaystyle 1-\exp\left\{-\int_s^{s+t} \lambda(u)du\right\} =
1-\exp\left\{-\int_0^t \lambda(s + v)dv\right\}.$  

If the function $ \lambda(t)$ is such that we can find a formula for the inverse $ F^{-1}_s$ for each $ s$, we can generate a random quantity $ X$ with the distribution $ F_s$ by using the inverse transform method. The algorithm, often called the integration method, can be summarized as follows:
Algorithm NHPP2 (Integration)
Step 1:
set $ T_0 = 0$
Step 2:
for $ i = 1,2,\ldots,n$ do
Step 2a:
generate a random variable $ U$ distributed uniformly on $ (0,1)$
Step 2b:
set $ T_i = T_{i-1} + F^{-1}_s(U)$

The third approach utilizes a generalization of the property used in the HPP2 algorithm. Given that $ N_t=n$, the $ n$ occurrence times $ T_1,T_2,\ldots,T_n$ of the non-homogeneous Poisson process have the same distributions as the order statistics corresponding to $ n$ independent random variables distributed on the interval $ (0,t]$, each with the common density function $ f(v)=\lambda(v)/\int_0^t \lambda(u)du$, where $ v\in (0,t]$. Hence, the arrival times of the NHPP on the interval $ (0,t]$ can be generated as follows:

Algorithm NHPP3
Step 1:
generate a Poisson random variable $ N$ with intensity $ \int_0^t\lambda(u)du$
Step 2:
generate $ N$ random variables $ V_i,i=1,2,\ldots N$ with density $ f(v)=\lambda(v)/\int_0^t \lambda(u)du$.
Step 3:
set $ (T_1,T_2,\ldots,T_N)=\textrm{sort}\{V_1,V_2,\ldots,V_N\}$.

The performance of the algorithm is highly dependent on the efficiency of the computer generator of random variables with density $ f(v)$. 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 $ \lambda(u)$ 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 $ \lambda (t)=a+b\cdot t$ are presented for the same value of parameter $ a$. Note, that the higher the value of parameter $ b$, the more pronounced is the increase in the intensity of the process. In the right panel realizations of a NHPP with periodic intensity $ \lambda(t)=a+b\cdot \cos(2\pi t)$ are illustrated, again for the same value of parameter $ a$. This time, for high values of parameter $ b$ the events exhibit a seasonal behavior. The process has periods of high activity (grouped around natural values of $ t$) and periods of low activity, where almost no jumps take place. Finally, we note that since in the non-homogeneous case the expected value $ \mathop{\textrm{E}}
(N_t)=\int_0^t\lambda(s)ds$, it is natural to define the premium function as $ c(t)=(1+\theta)\mu\int_0^t\lambda (s)ds$.


14.2.3 Mixed Poisson Process

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 $ \textrm{Var} (N_t)/\mathop{\textrm{E}} (N_t)$, 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 $ \{N_t\}$ is given by a mixture of Poisson processes (Rolski et al.; 1999). This means that, conditioning on an extrinsic random variable $ \Lambda$ (called a structure variable), the process $ \{N_t\}$ behaves like a homogeneous Poisson process. Since for each $ t$ the claim numbers $ \{N_t\}$ up to time $ t$ are Poisson variates with intensity $ \Lambda t$, it is now reasonable to consider the premium function of the form $ c(t)=(1+\theta)\mu\Lambda t$.

The process can be generated in the following way: first a realization of a non-negative random variable $ \Lambda$ is generated and, conditioned upon its realization, $ \{N_t\}$ 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:

Algorithm MPP1
Step 1:
generate a realization $ \lambda $ of the random intensity $ \Lambda$
Step 2:
set $ T_0 = 0$
Step 3:
for $ i = 1,2,\ldots,n$ do
Step 3a:
generate an exponential random variable $ E$ with intensity $ \lambda $
Step 3b:
set $ T_i = T_{i-1} + E$


14.2.4 Cox Process

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 $ \{\Lambda(t)\}$ is used to generate another process $ \{N_t\}$ by acting as its intensity. That is, $ \{N_t\}$ is a Poisson process conditional on $ \{\Lambda(t)\}$ which itself is a stochastic process. If $ \{\Lambda(t)\}$ is deterministic, then $ \{N_t\}$ is a non-homogeneous Poisson process. If $ \Lambda(t)=\Lambda$ for some positive random variable $ \Lambda$, then $ \{N_t\}$ 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 $ c(t)=(1+\theta)\mu\int_0^t\Lambda (s)ds$.

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 $ \{\Lambda(t)\}$ is generated and, conditioned upon its realization, $ \{N_t\}$ 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:

Algorithm CP1
Step 1:
generate a realization $ \lambda(t)$ of the intensity process $ \{\Lambda(t)\}$ for a sufficiently large time period
Step 2:
set $ \overline{\lambda} = \max\left\{\lambda(t) \right\}$
Step 3:
set $ T_0 = 0$ and $ T^* = 0$
Step 4:
for $ i = 1,2,\ldots,n$ do
Step 4a:
generate an exponential random variable $ E$ with intensity $ \overline{\lambda}$
Step 4b:
set $ T^* = T^* + E$
Step 4c:
generate a random variable $ U$ distributed uniformly on $ (0,1)$
Step 4d:
if $ U>\lambda(T^*)/\overline{\lambda}$ then return to step 4a ( $ \rightarrow$ reject the arrival time) else set $ T_i = T^*$ ( $ \rightarrow$ accept the arrival time)


14.2.5 Renewal Process

Generalizing the homogeneous Poisson process we come to the point where instead of making $ \lambda $ non-constant, we can make a variety of different distributional assumptions on the sequence of waiting times $ \{W_1,W_2,\ldots\}$ of the claim arrival point process $ \{N_t\}$. In some particular cases it might be useful to assume that the sequence is generated by a renewal process, i.e. the random variables $ W_i$ 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:

Algorithm RP1
Step 1:
set $ T_0 = 0$
Step 2:
for $ i = 1,2,\ldots,n$ do
Step 2a:
generate a random variable $ X$ with an assumed distribution function $ F$
Step 2b:
set $ T_i = T_{i-1} + X$

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 $ \{N_t\}$ be a renewal process and assume that $ W_1$ has finite mean $ 1/\lambda$. Then the premium function is defined in a natural way as $ c(t)=(1+\theta)\mu\lambda t$, like for the homogeneous Poisson process.