| The Homogeneous Poisson Process, the Intensity Function, the Cramer-Lundberg Model |
| Written by theoretic |
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The most popular Poisson process corresponds to the case of a linear mean value function /л: /x(t) = At, t > 0 , for some A > 0. A process with such a mean value function is said to be homogeneous, inhomogeneous otherwise. The quantity A is the intensity or rate of the homogeneous Poisson process. If A = 1, N is called standard homogeneous Poisson process. More generally, we say that N has an intensity function or rate function A if /x is absolutely continuous, i.e., for any s < t the increment n(s,t] has representation t (j,(s,t]= / X(y) dy , s<t, s for some non-negative measurable function A. A particular consequence is that /x is a continuous function. We mentioned that /л can be interpreted as operational time or inner clock of the Poisson process. If N is homogeneous, time evolves linearly: n(s,t] = /x(s + h,t + h] for any h > 0 and 0 < s < t < 00. Intuitively, this means that claims arrive roughly uniformly over time. We will see later, in Section 2.1.6, that this intuition is supported by the so-called order statistics property of a Poisson process. If N has non-constant intensity function A time “slows down” or “speeds up” according to the magnitude of A(t). In Figure 2.1.2 we illustrate this effect for different choices of A. In an insurance context, non-constant A may refer to seasonal effects or trends. For example, in Denmark more car accidents happen in winter than in summer due to bad weather conditions. Trends can, for example, refer to an increasing frequency of (in particular, large) claims over the last few years. Such an effect has been observed in windstorm insurance in Europe and is sometimes mentioned in the context of climate change. Table 3.2.18 contains the largest insurance losses occurring in the period 1970-2002: it is obvious that the arrivals of the largest claim sizes cluster towards the end of this time period. We also refer to Section 2.1.7 for an illustration of seasonal and trend effects in a real-life claim arrival sequence. |