The corresponding sample path
User Rating: / 0
PoorBest 
Written by theoretic   
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.
Figure 1.0.1 A sample path of the claim arrival process N (left) and of the corresponding total claim amount process S (right). Mind the difference of the jump sizes!
• Find sufficiently realistic, but simple,2 probabilistic models for S and N. This means that we have to specify the distribution of the claim sizes Xi and to introduce models for the claim arrival times Тi. The discrepancy between “realistic” and “simple” models is closely related to the question to which extent a mathematical model can describe the complicated dynamics of an insurance portfolio without being mathematically intractable.
• Determine the theoretical properties of the stochastic processes S and N. Among other things, we are interested in the distributions of S and N, their distributional characteristics such as the moments, the variance and the dependence structure. We will study the asymptotic behavior of N(t) and S(t) for large t and the average behavior of N and S in the interval [0,t]. To be more specific, we will give conditions under which the strong law of large numbers and the central limit theorem hold for S and N.
• Give simulation procedures for the processes N and S. Simulation methods have become more and more popular over the last few years. In many cases they have replaced rigorous probabilistic and/or statistical methods. The increasing power of modern computers allows one to simulate various scenarios of possible situations an insurance business might have to face in the future. This does not mean that no theory is needed any more. On the contrary, simulation generally must be based on probabilistic models for N and S; the simulation procedure itself must exploit the theoretical properties of the processes to be simulated.
• Based on the theoretical properties of N and S, give advice how to choose a premium in order to cover the claims in the portfolio, how to build reserves, how to price insurance products, etc.