| The Cramer-Lundberg model |
| Written by theoretic |
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The homogeneous Poisson process plays a major role in insurance mathematics. If we specify the claim number process as a homogeneous Poisson process, the resulting model which combines claim sizes and claim arrivals is called Cramer-Lundberg model : • Claims happen at the arrival times 0 < T\ < T2 < of a homogeneous Poisson process N(t) = #{« > 1 : Tj < t}, t > 0. • The ith claim arriving at time Ti causes the claim size X$. The sequence (Xi) constitutes an iid sequence of non-negative random variables. • The sequences (Ti) and (Xi) are independent. In particular, N and (Xi) are independent. The total claim amount process S in the Cramer-Lundberg model is also called a compound Poisson process. The Cramer-Lundberg model is one of the most popular and useful models in non-life insurance mathematics. Despite its simplicity it describes some of the essential features of the total claim amount process which is observed in reality. We mention in passing that the total claim amount process S in the Cramer-Lundberg setting is a process with independent and stationary in crements, starts at zero and has cadlag sample paths. It is another important example of a Levy process. Try to show these properties! □ Comments The reader who wants to learn about Levy processes is referred to Sato’s monograph [71]. For applications of Levy processes in different areas, see the recent collection of papers edited by Barndorff-Nielsen et al. [9]. Rogers and Williams [66] can be recommended as an introduction to Brownian motion, its properties and related topics such as stochastic differential equations. For an elementary introduction, see Mikosch [57]. |