Risk theory for the compound Poisson process that is perturbed by diffusion. (English) Zbl 0723.62065

Summary: The classical model of collective risk theory is extended in that a diffusion process is added to the compound Poisson process. It is shown that the probabilities of ruin (by oscillation or by a claim) satisfy certain defective renewal equations. The convolution formula for the probability of ruin is derived and interpreted in terms of the record highs of the aggregate loss process. If the distributions of the individual claim amounts are combinations of exponentials, the probabilities of ruin can be calculated in a transparent fashion. Finally, the role of the adjustment coefficient (for example, for the asymptotic formulas) is explained.


62P05 Applications of statistics to actuarial sciences and financial mathematics
45J05 Integro-ordinary differential equations
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