In the first half of the diploma, we define the compound Poisson distribution and derive the form of its generating functions. We discuss its connection with general distributions and derive the Panjer recursion scheme. We then define the compound Poisson process and show some basic properties such as the independence and stationarity of increments. We derive some results that follow from a space-time decomposition of the compound Poisson process. In the second half of the diploma, we discuss the application of the compound Poisson process in the Cramér-Lundberg model. We define the probability of ruin and survival and express the latter as a defective renewal equation. We prove the Lundberg inequality and discuss the asymptotic behavior of the probability of ruin when claims are modeled with light-tailed and heavy-tailed distributions. We practically demonstrate the behavior
of the probability of ruin by repeatedly simulating the risk process.
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