The Total Claim Amount

In Chapter 2 we learned about three of the most prominent claim number processes, N: the Poisson process in Section 2.1, the renewal process in Section 2.2, and the mixed Poisson process in Section 2.3. In this section we take a closer look at the total claim amount process, as introduced on p. 4:

$$ |S(t) = \sum\limits_{i = 1}^{N(t)} {X_i ,t \ge 0,}$$

((3.0.1))

where the claim number process N is independent of the iid claim size sequence (Xi). We also assume that Xi > 0 a.s. Depending on the choice of the process N, we get different models for the process S. In Example 2.1.3 we introduced the Craméer-Lundberg model as that particular case of model (3.0.1) when N is a homogeneous Poisson process. Another prominent model for S is called renewal or Sparre-Anderson model; it is model (3.0.1) when N is a renewal process.