zbMATH — the first resource for mathematics

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
Full Text: DOI
[1] Beekman, J.A., Two stochastic processes, (1974), Almqvist & Wiksell Stockholm · Zbl 0137.35601
[2] Bowers, N.J.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; Nesbitt, C.J., Actuarial mathematics, (1986), Society of Actuaries Itasca, IL
[3] Chan, B., Ruin probability for translated combination of exponential claims, ASTIN bulletin, 20, 113-114, (1990)
[4] Dufresne, F.; Gerber, H.U., Three methods to calculate the probability of ruin, ASTIN bulletin, 19, 71-90, (1989)
[5] Feller, W.S., Introduction to probability theory and its applications, II, (1966), Wiley New York · Zbl 0138.10207
[6] Gerber, H.U., An extension of the renewal equation and its application in the collective theory of risk, Scandinavian actuarial journal, 205-210, (1970) · Zbl 0229.60062
[7] Täcklind, S., Sur le risque de ruine dans des jeux inéquitables, Scandinavian actuarial journal, 1-42, (1942) · JFM 68.0276.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.