Accumulated claims and collective risk in insurance: Higher order asymptotic approximations

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Abstract
A stochastic process modelling the reserves of an insurance company is introduced. After a slight modification, the model can be reinterpreted for the analysis of the amount of claims against an insurance company as well. The flexibility this model provides may prove useful when approaching real problems in the insurance industry. With some simplifying assumptions, it becomes possible to compute the probability distribution of the claims at each time epoch. The epression for the exact distribution becomes complicated after a long period of time has elapsed. Hence the need for asymptotic expansions arises. Conditions are found under which a higher order asymptotic expression is valid. The Berry-Esséen approach is used in the proof of this result. There are problems with establishing an expansion for densities. These are briefly discussed. A result is provided for approximating the tails of the distribution F(x; t). The result constrains the size of |x|, as it must be compared to t. Finally, an equation for the probability of ruin is derived. The model used here allows, among other things, for uncertainty in the premium income of the company. This aspect adds realism to the model, since the concept of uncertainty is essential when analyzing the market.