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Ruin probabilities in the compound Markov binomial model. (English) Zbl 1092.91040

This paper deals with a compound Markov binomial model which is an extension of the compound binomial model. In the compound binomial model, the time is measured in discrete units \(0,1,2,\ldots\) and the surplus process is defined by \(U_{k}=u+k-S_{k}\) for \(k=0,1,2\ldots\) since premiums are payable at rate of 1 per time unit, \(U_0=u\), where \(u\in\{0,1,2,\ldots\}\). The total claim amount over \(k\) periods \(S_{k}\) is defined by \[ S_{k}=\begin{cases} \sum_{j=1}^{M_{k}}X_{j}, & M_{k}>0,\\ 0, & M_{k}=0,\end{cases} \] where \(M_{k}\) represents the total number of claims over \(k\) periods, \(M_{k}=I_1+\ldots+I_{k}\). It is assumed that \(\{I_{k},k=0,1,2,\ldots\}\) is a stationary homogeneous Markov chain with the state space \(\{0,1\}\). The authors present properties of the Markov Bernoulli and Markov binomial models and then examine the computation of the finite and infinite-time ruin probabilities. The impact of the dependence on the ruin probability is shown and the Lundberg exponential bound for such probability is derived. Some numerical examples are presented.

MSC:

91B30 Risk theory, insurance (MSC2010)
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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[1] Cossette H., Insurance: Mathematics and Economics 26 pp 133– (2000) · Zbl 1103.91358
[2] Denuit, M., Genest, C. & Marceau, E. (2002). Criteria for the stochastic ordering of random sums, with actuarial applications.Scandinavian Actuarial Journal, 3–17. · Zbl 1003.60022
[3] De Vylder, F. (1996).Advanced risk theory: a self-contained introduction. Editions de l’Université de Bruxelles, Bruxelles.
[4] De Vylder, F. & Marceau, E. (1996). Classical numerical ruin probabilities.Scandinavian Actuarial Journal, 109–123. · Zbl 0880.62108
[5] Dickson D. C. M., ASTIN Bulletin 24 pp 33– (1994)
[6] Dickson D. C. M., ASTIN Bulletin 25 pp 153– (1995)
[7] Gerber, E. (1979).An introduction to mathematical risk theory. S.S. Huebner Foundation for Insurance Education, Philadelphia.
[8] Gerber E., ASTIN Bulletin 18 pp 161– (1988)
[9] Gerber E., Insurance: Mathematics nad Economics 7 pp 15– (1988) · Zbl 0657.62121
[10] Kaas, R., van Heerwaarden, A. E. & Goovaerts, M. J. (1994).Ordering of actuarial risks. CAIRE Education Series 1, Brussels. · Zbl 0683.62060
[11] Karlin, S. & Taylor, H. (1975).A first course in stochastic processes 2nd edition. Academic Press, New York. · Zbl 0315.60016
[12] Klugman, S. A., Panjer, H. H. & Willmot, G. E. (1998).Loss models: from data to decisions. Wiley, New York. · Zbl 0905.62104
[13] Michel R., Insurance: Mathematics and Economics 8 pp 149– (1989) · Zbl 0676.62085
[14] Rolski, T., Schmidli, H., Schmidt, V. & Teugels, J. (1999).Stochastic processes for insurance and finance. Wiley, New York. · Zbl 0940.60005
[15] Serfozo R., The Annals of Probability 14 pp 1391– (1986) · Zbl 0604.60016
[16] Shiryayev, A. N. (1996).Probability. Springer-Verlag, New York.
[17] Shiu E., ASTIN Bulletin 19 pp 179– (1989)
[18] Vellaisamy P., Statistics and Probability Letters 41 pp 179– (1999) · Zbl 0921.60020
[19] Wang Y. H., Journal of Applied Probability 18 pp 937– (1981) · Zbl 0475.60050
[20] Willmot G. E., Insurance: Mathematics and Economics 12 pp 133– (1993) · Zbl 0778.62099
[21] Willmot, G. E. & Lin, S. (2001).Lundberg approximations for compound distributions with insurance applications. Lecture notes in statistics, Springer-Verlag, New-York.
[22] Yuen K. C., Insurance: Mathematics and Economics 29 pp 47– (2001) · Zbl 1074.91032
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