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Phase-type distributions and risk processes with state-dependent premiums. (English) Zbl 0876.62089
Summary: Consider a risk reserve process with initial reserve \(u\), Poisson arrivals, premium rule \(p(r)\) depending on the current reserve \(r\) and claim size distribution which is phase-type in the sense of M. F. Neuts [see “Matrix-geometric solutions in stochastic models. An algorithmic approach” (1981; Zbl 0469.60002)]. It is shown that the ruin probabilities \(\psi(u)\) can be expressed as the solution of a finite set of differential equations, and similar results are obtained for the case where the process evolves in a Markovian environment (e.g., a numerical example of a stochastic interest rate is presented). Further, an explicit formula for \(\psi(u)\) is presented for the case where \(p(r)\) is a two-step function. By duality, the results apply also to the stationary distribution of storage processes with the same input and release rate \(p(r)\) at content \(r\).

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
62M99 Inference from stochastic processes
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[1] Sparre Andersen E., Transactions XVth International Congress of Actuaries, New York, II pp 219– (1957)
[2] Asmussen S., Applied Probability and Queues (1987)
[3] Asmussen S., Ruin Probabilities (1995)
[4] Asmussen S., Does Markov-modulation increase the risk? (1994)
[5] Asmussen S., Ruin Probabilities via local adjustment coefficients (1995) · Zbl 0834.60099
[6] DOI: 10.2307/1427367 · Zbl 0657.60111
[7] DOI: 10.1016/0167-6687(92)90058-J · Zbl 0748.62058
[8] Baker C. T. H., The Numerical Solution of Integral Equations (1977)
[9] Bladt M., Ph.D. thesis (1993)
[10] Cramer H., On the Mathematical Theory of Risk (1930)
[11] Emanuel D. C., Scand. Actuarial J. pp 37– (1975)
[12] Feller W., An Introduction to Probability Theory and its Applications. Vol. II, 2. ed. (1971) · Zbl 0219.60003
[13] Gerber H., Mitt. Verein Schweitz. Versich. Math. 71 pp 63– (1971)
[14] Gerber H., An Introduction to the Mathematical Risk Theory (1979) · Zbl 0431.62066
[15] Graham A., Kronecker Products and Matrix Calculus with Applications (1981) · Zbl 0497.26005
[16] DOI: 10.1007/978-1-4613-9058-9
[17] DOI: 10.1016/0304-4149(77)90051-5 · Zbl 0361.60053
[18] Neuts M. F., Matrix-Geometric Solutions in Stochastic Models (1981) · Zbl 0469.60002
[19] Schock Petersen S., Scand. Actuarial J. pp 147– (1990)
[20] Taylor G. C., Scand. Act. J. pp 57– (1980) · Zbl 0426.62069
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