zbMATH — the first resource for mathematics

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 \(r\).

62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
62M99 Inference from stochastic processes
Full Text: DOI
[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
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.