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Risk theory in an economic environment and Markov processes. (English) Zbl 0816.90042
Summary: Two models of the collective theory of risk, one introduced by H. U. Gerber [Schweiz. Verein. Versicherungsmath. Mitt. 71, 63-70 (1971)] and the other by A. Dassios and P. Embrechts [Commun. Stat., Stochastic Models 5, No. 2, 181-217 (1989; Zbl 0676.62083)], where borrowing is allowed, are extended with the possibility of investment above a certin level. An application of Davis’ method of piecewise deterministic Markov processes [see M. H. A. Davis, J. R. Stat. Soc., Ser. B 46, 353-388 (1984; Zbl 0565.60070)] yields the Laplace transform of the ruin probabilities as well as a ‘Lundberg exponent’ for the model. For the case of equal interest rates for invested and borrowed money an explicit inversion formula is given.

MSC:
91B30 Risk theory, insurance (MSC2010)
60J99 Markov processes
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