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Computational methods in risk theory: a matrix-algorithmic approach. (English) Zbl 0748.62058
The paper discusses and shows the probabilities of ruin, $$\psi(u)$$, obtained from fitting a selection of conditional claim amount distributions by phase-type distributions. The matrix-algorithmic method, initially due to Neuts, and associated with the latter distributions, when carried over to a continuous-state setting, provides computationally tractable solutions for $$\psi(u)$$; the example of the compound Poisson distribution is particularly elegant as evidenced by the application to hyperexponential claims. Other examples illustrated are for the Sparre Andersen process, Markov modulated arrivals, and periodic Poisson processes.
Reviewer: G.Lord (Princeton)

##### MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics 65C99 Probabilistic methods, stochastic differential equations 91B30 Risk theory, insurance (MSC2010)
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##### References:
 [1] Afanas’eva, L.L., On periodic distribution of waiting-time process, (), 1-20, Lecture Notes in Math. 1155 · Zbl 0599.60086 [2] Andersen, E.Sparre, On the collective theory of risk in case of contagion between the claims, Transactions xvth international congress of actuaries., New York, II, 219-229, (1957) [3] Asmussen, S., Approximations for the probability of ruin within finite time, Scand. actuarial. J., 31-57, (1984) · Zbl 0568.62092 [4] Asmussen, S., Approximations for the probability of ruin within finite time, Scand. actuarial. J., 64, (1985) [5] Asmussen, S., Applied probability and queues, (1987), Wiley Chichester · Zbl 0624.60098 [6] Asmussen, S., Risk theory in a Markovian environment, Scand. actuarial J., 69-100, (1989) · Zbl 0684.62073 [7] Asmussen, S., Ladder heights and the Markov-modulated M / G/1 queue, Stoch. proc. appl., 39, 2, (1991) · Zbl 0734.60091 [8] Asmussen, S., Phase-type representations in random walk and queueing problems, The annals of probability, 20, (1992), forthcoming · Zbl 0755.60049 [9] Asmussen, S., Fundamentals of ruin probability theory, (1992), Incomplete book manuscript [10] Asmussen, S.; Nerman, O., The EM algoritm for phase-type distributions viewed as incompletely observed exponential families, in preparation, (), 335-346, (1991) [11] Asmussen, S.; Rolski, T., Risk theory in a periodic environment: the cramér—lundberg approximation and Lundberg’s inequality, (1991), Submitted for publication [12] Asmussen, S.; Thorisson, H., A Markov chain approach to periodic queues, J. appl. probab., 24, 215-225, (1987) · Zbl 0623.60116 [13] Benckert, L.-G.; Jung, J., Statistical models of claim distributions in fire insurance, ASTIN bull., VIII, 1-25, (1974) [14] Björk, T.; Grandell, J., Exponential inequalities for ruin probabilities in the Cox case, Scand. actuarial J., 77-111, (1988) · Zbl 0668.62072 [15] Bobbio, A.; Cumani, A., ML estimation of the parameters of a PH distribution in triangular canonical form, (1990), Technical report [16] Burman, D.Y.; Smith, D.R., An asymptotic analysis of a queueing system with Markov-modulated arrivals, Opns. res., 34, 105-119, (1986) · Zbl 0593.90031 [17] Bux, W.; Herzog, U., The phase concept: approximation of measured data and performance analysis, (), 23-38 [18] Dassios, A.; Embrechts, P., Martingales and insurance risk, Stochastic models, 5, 181-218, (1989) · Zbl 0676.62083 [19] Embrechts, P.; Veraverbeke, N., Estimates of the probability of ruin with special emphasis on the possibility of large claims, Insurance: mathematics and economics, 1, 55-72, (1982) · Zbl 0518.62083 [20] Falin, G.I., Periodic queues in heavy traffic, Adv. appl. prob., 21, 485-487, (1989) · Zbl 0668.60082 [21] Gerber, H.U., An introduction to mathematical risk theory, (1979), S.S. Huebner Foundation Monographs, University of Pennsylvania Philadelphia, PA · Zbl 0431.62066 [22] Graham, A., Kronecker products and matrix calculus with applications, (1981), Ellis Horwood Chichester · Zbl 0497.26005 [23] Grandell, J., A class of approximations of ruin probabilities, Scand. actuarial J., 37-52, (1977) · Zbl 0384.60057 [24] Harrison, J.M.; Lemoine, A.J., Limit theorems for periodic queues, J. appl. probab., 14, 566-576, (1977) · Zbl 0372.60127 [25] Janssen, J., Some transient results on the M / SM /1 special semi-Markov model in risk and queueing theories, ASTIN bull., 11, 41-51, (1980) [26] Janssen, J.; Reinhard, J.M., Probabilités de ruine pour une classe de modeles de risque semi-markoviens, ASTIN bull., 15, 123-133, (1985) [27] Johnson, M.A.; Taaffe, M.R., Matching moments to phase distributions: mixtures of Erlang distributions of common order, Stochastic models, 6, 259-281, (1989) · Zbl 0684.60012 [28] Lemoine, A.J., On queues with periodic Poisson input, J. appl. prob., 18, 889-900, (1981) · Zbl 0472.60084 [29] Lemoine, A.J., Waiting time and workload in queues with periodic Poisson input, J. appl. probab., 26, 390-397, (1989) · Zbl 0683.60069 [30] Lucantoni, D.M., New results on the single server queue with a batch Markovian arrival process, Stoch. models, 6, (1990) [31] Moler, C.; Van Loan, C., Nineteen dubious ways to compute the exponential of a matrix, SIAM review, 20, 801-836, (1978) · Zbl 0395.65012 [32] Neuts, M.F., Matrix-geometric solutions in stochastic models, (1981), Johns Hopkins University Press Baltimore, MD · Zbl 0469.60002 [33] Neuts, M.F., Structured stochastic matrices of the M / G / 1 type and their applications, (1989), Marcel Dekker New York [34] Press, W.H.; Flannery, B.P.; Teukalsky, S.A.; Valtering, W.T., Numerical recipes. the art of scientific computing, (1988), Cambridge University Press Cambridge [35] Ramaswami, V., Nonlinear matrix equations in applied probability, SIAM review, 30, 256-263, (1986) · Zbl 0642.65033 [36] Ramaswami, V., From the matrix-geometric to the matrix-exponential, Queueing systems, (1990) · Zbl 0702.60084 [37] Regterschot, G.J.K.; de Smit, J.H.A., The queue M / G / 1 with Markov-modulated arrivals and services, Math. oper. res., 11, 456-483, (1986) · Zbl 0619.60093 [38] Reinhard, J.M., On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment, ASTIN bull., XIV, 23-43, (1984) [39] Rolski, T., Queues with non-stationary input stream: Ross’s conjecture, Adv. appl. prob., 13, 603-618, (1981) · Zbl 0462.60090 [40] Rolski, T., Approximation of periodic queues, J. appl. probab., 19, 691-707, (1987) · Zbl 0631.60092 [41] Rolski, T., Relationships between characteristics in periodic Poisson queues, Queueing systems, 4, 17-26, (1989) · Zbl 0676.60087 [42] Rolski, T., Queues with nonstationary inputs, Queueing systems, 5, 113-130, (1990) · Zbl 0687.60082 [43] Sengupta, B., Markov processes whose steady-state distribution is matrix-exponential with an application to the GI / PH / 1 queue, Adv. appl. probab., 21, 159-180, (1989) · Zbl 0672.60090 [44] Sengupta, B., The semi-Markovian queue: theory and applications, Stochastic models, (1990) · Zbl 0699.60090 [45] Thorin, O., Some remarks on the ruin problem in case the epochs of claims form a renewal process, Scand. actuarial. J., 29-50, (1970) · Zbl 0218.60082 [46] Thorin, O., Further remarks on the ruin problem in case the epochs of claims for a renewal process, Scand. actuarial. J., 14-38, (1971) · Zbl 0246.62095 [47] Thorin, O., Further remarks on the ruin problem in case the epochs of claims for a renewal process, Scand. actuarial. J., 121-142, (1971) · Zbl 0283.62096 [48] Thorin, O., On the asymptotic behaviour of the ruin probability for an infinite period when epochs of claims form a renewal process, Scand. actuarial. J., 81-99, (1974) · Zbl 0288.60082 [49] Thorin, O., Stationarity aspects of the sparre Andersen risk process and the corresponding ruin probabilities, Scand. actuarial. J., 87-98, (1975) · Zbl 0315.60050 [50] Thorin, O.; Wikstad, N., Numerical evaluation of ruin probabilities, ASTIN bull., VII, 137-153, (1973) [51] Thorin, O.; Wikstad, N., Numerical evaluation of ruin probabilities when the claim distribution is lognormal, ASTIN bull., VIII, 231-245, (1974)
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