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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.

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