| Poisson process |
| Written by theoretic |
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This concerns the renewal process which is considered in Section 2.2. It allows for more flexibility in choosing the distribution of the inter-arrival times Ti —Ti-\. But one has to pay a price: in contrast to the Poisson process when N(t) has a Poisson distribution for every t, this property is in general not valid for a renewal process. Moreover, the distribution of N(t) is in general not known. Nevertheless, the study of the renewal process has led to a strong mathematical theory, the so-called renewal theory, which allows one to make quite precise statements about the expected claim number EN(t) for large t. We sketch renewal theory in Section 2.2.2 and explain what its purpose is without giving all mathematical details, which would be beyond the scope of this text. We will see in Section 4.2.2 on ruin probabilities that the so-called renewal equation is a very powerful tool which gives us a hand on measuring the probability of ruin in an insurance portfolio. A third model for the claim number process N is considered in Section 2.3: the mixed Poisson process. It is another modification of the Poisson process. By randomization of the parameters of a Poisson process (“mixing”) one obtains a class of processes which exhibit a much larger variety of sample paths than for the Poisson or the renewal processes. We will see that the mixed Poisson process has some distributional properties which completely differ from the Poisson process. After the extensive study of the claim number process we focus in Chapter 3 on the theoretical properties of the total claim amount process S. We start in Section 3.1 with a description of the order of magnitude of S(t). Results include the mean and the variance of S(t) (Section 3.1.1) and asymptotic properties such as the strong law of large numbers and the central limit theorem for S(t) as t —> oo (Section 3.1.2). We also discuss classical premium calculation principles (Section 3.1.3) which are rules of thumb for how large the premium in a portfolio should be in order to avoid ruin. These principles are consequences of the theoretical results on the growth of S(t) for large t. In Section 3.2 we hint at realistic claim size distributions. In particular, we focus on heavy-tailed claim size distributions and study some of their theoretical properties. Distributions with regularly varying tails and subexponential distributions are introduced as the natural classes of distributions which are capable of describing large claim sizes. Section 3.3 continues with a study of the distributional characteristics of S(t). We show some nice closure properties which certain total claim amount models (“mixture distributions”) obey; see Section 3.3.1. We also show the surprising result that a disjoint decomposition of time and/or claim size space yields independent total claim amounts on the different pieces of the partition; see Section 3.3.2. Then various exact (numerical; see Section 3.3.3) and approximate (Monte Carlo, bootstrap, central limit theorem based; see Section 3.3.4) methods for determining the distribution of S(t), their advantages and drawbacks are discussed. Finally, in Section 3.4 we give an introduction to reinsurance treaties and show the link to previous theory. |