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In 1903 the Swedish actuary Filip Lundberg [55] laid the foundations of modern risk theory. Risk theory is a synonym for non-life insurance mathematics, which deals with the modeling of claims that arrive in an insurance business and which gives advice on how much premium has to be charged in order to avoid bankruptcy (ruin) of the insurance company.
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The iid property of the claim sizes, Xi, reflects the fact that there is a homogeneous probabilistic structure in the portfolio. The assumption that claim sizes and claim times be independent is very natural from an intuitive point of view. But the independence of claim sizes and claim arrivals also makes the life of the mathematician much easier, i.e., this assumption is made for mathematical convenience and tractability of the model. Now we can define the claim number process
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The corresponding sample path |
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In Figure 1.0.1 we see a sample path of the process N and the corresponding sample path of the compound sum process S. Both paths jump at the same times Ti: by 1 for N and by Xi for S.
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Although statistical inference on the processes S and N is utterly important for the insurance business, we do not address this aspect in a rigorous way. The statistical analysis of insurance data is not different from standard statistical methods which have been developed for iid data and for counting processes.
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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.
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A major building block of classical risk theory is devoted to the probability of ruin; see Chapter 4. It is a global measure of the risk one encounters in a portfolio over a long time horizon. We deal with the classical small claim case and give the celebrated estimates of Cram´er and Lundberg (Sections 4.2.1 and 4.2.2).
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In this section we consider the most common claim number process: the Poisson process. It has very desirable theoretical properties. For example, one can derive its finite-dimensional distributions explicitly.
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A stochastic process N = (N(t))t>o is said to be a Poisson process if the
following conditions hold:
(1) The process starts at zero: N(0) = 0 a.s.
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The Homogeneous Poisson Process, the Intensity Function, the Cramer-Lundberg Model |
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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.
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