Abstract
In this paper semi-Markov reward models are presented. Higher moments of the reward process is presented for the first time applied to in time non-homogeneous semi-Markov insurance problems. Also an example is presented based on real disability data. Different algorithmic approaches to solve the problem is described.
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Y. Balcer, and I. Sahin, “Pension accumulation as a semi-Markov reward process, with applications to pension reform.” In J. Janssen (eds), Semi-Markov Models, Plenum: New York, 1986.
A. Blasi, J. Janssen, and R. Manca, “Generalized discrete time homogeneous stochastic annuities and multi-state insurance model,” In Proceedings of IME 2004, Rome, Italy, 2004.
CMIR12, (Continuous Mortality Investigation Report, no. 12), “The analysis of permanent health insurance data,” The Institute of Actuaries and the Faculty of Actuaries, 1991.
R. Consael, and J. Sonnenscheim, “Theorie mathematique des assurances des personnes. Modèle markovien,” Mitteilungen der Vereinigung schweizerischer Versincherungsmathematik vol.78 pp. 75–93, 1978.
E. Çinlar, “Markov renewal theory,” Advances in Applied Probability vol. 1 pp. 123–187, 1969.
R. De Dominicis, and R. Manca, “An algorithmic approach to non-homogeneous semi-Markov processes,” Communications in Statistics. Simulation and Computation vol.13 pp. 113–127, 1984.
R. De Dominicis, and R. Manca, “Some new results on the transient behaviour of semi-Markov reward processes,” Methods of Operations Research vol. 54, 1986.
R. De Dominicis, R. Manca, and L. Granata, “The dynamics of pension funds in a stochastic environment,” Scandinavian Actuarial Journal 1991.
S. Haberman, and E. Pitacco, Actuarial Models for Disability Insurance, Chapman and Hall: London, UK, 1999.
J. M. Hoem, “Markov chain models in life insurance,” Blätter der Deutschen Gesellschaft für Versincherungsmathematik vol. 9 pp. 91–107, 1969.
J. M. Hoem, “Inhomogeneous semi-Markov processes, select actuarial tables, and duration-dependence in demography.” In T.N.E. Greville (ed.), Population, Dynamics, pp. 251–296, Academic: NY, 1972.
J. M. Hoem, “The versatility of the Markov chain as a tool in the mathematics of life insurance,” Transactions of the 23rd Congress of Actuaries vol. R pp. 141–202, 1988.
A. Iosifescu Manu, “Non homogeneous semi-Markov processes,” Stud. Lere. Mat. vol. 24 pp. 529–533, 1972.
J. Janssen, “Application des processus semi-markoviens à un probléme d’invalidité,” Bulletin de l’Association Royale des Actuaries Belges vol. 63 pp. 35–52, 1966.
J. Janssen, and R. De Dominicis, “Finite non-homogeneous semi-Markov processes,” Insurance: Mathematics and Economics vol. 3 pp. 157–165, 1984.
J. Janssen, and R. Manca, “A realistic non-homogeneous stochastic pension funds model on scenario basis,” Scandinavian Actuarial Journal vol. 2 pp. 113–137, 1997.
J. Janssen, and R. Manca, “General actuarial models in a semi-Markov environment,” In Proceedings of ICA Cancun 2002, Mexico, 2002.
J. Janssen, and R. Manca, “Multi-state insurance model description by means of continuous time homogeneous semi-Markov reward processes,” Mimeo, 2003.
J. Janssen, and R. Manca, “Discrete time non-homogeneous semi-Markov reward processes, generalized stochastic annuities and multi-state insurance model,” In Proceedings of XXVIII AMASES, Modena, 2004.
J. Janssen, and R. Manca, Applied Semi-Markov Processes, Springer, Berlin Heidelberg New York, 2006.
J. Janssen, R. Manca, and E. Volpe di Prignano, “Continuous time non homogeneous semi-Markov reward processes and multi-state insurance application,” In Proceedings of IME 2004, 2004.
R. Manca, D. Silvestrov, F. Stenberg, “Homogeneous Backward Semi-Markov Reward Models for Insurance Contracts,” In Procedings Applied Stochastic Models and Data Analysis 2005, ISBN 2-908849-15-1, 2005.
C. M. Moller, “Numerical evaluation of Markov transition probabilities based on the discretized product integral,” Scandinavian Actuarial Journal pp. 76–87, 1992.
R. Norberg, “Identities for present values of life insurance benefits,” Scandinavian Actuarial Journal pp. 100–106, 1993.
I. Sahin, and Y. Balcer, “Stochastic models for a pensionable service,” Operational Research vol. 27 pp. 888–903, 1979.
F. Stenberg, R. Manca, and D. Silvestrov, “Discrete time backward semi-Markov reward processes and an application to disability insurance problems,” Research Report 2005-1 Department of Mathematics and Physics, Mälardalen University, Sweden, 1–44, ISSN 1494-4978, 2005.
F. Stenberg, R. Manca, and D. Silvestrov, “Discrete time homogenous backward semi-Markov reward processes and an application to disability insurance,” 2005.
H. Waters, “An approach for the study of multiple state models,” Journal of the Institute of Actuaries vol. 116 pp. 611–624, 1984.
H. Wolthuis, “Actuarial equivalence,” Insurance: Mathematics and Economics vol. 15 pp. 163–179, 1994.
H. Wolthuis, Life Insurance Mathematics (The Markovian Model) IAE, Universiteit van Amsterdam, Amsterdam II edition, 2003.
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This work is partly supported by the Knowledge Foundation and Sparbankens Stiftelse Nya. The authors would like to thank the anonymous referee.
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Stenberg, F., Manca, R. & Silvestrov, D. An Algorithmic Approach to Discrete Time Non-homogeneous Backward Semi-Markov Reward Processes with an Application to Disability Insurance. Methodol Comput Appl Probab 9, 497–519 (2007). https://doi.org/10.1007/s11009-006-9012-4
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DOI: https://doi.org/10.1007/s11009-006-9012-4