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Compound binomial risk model in a Markovian environment. (English) Zbl 1079.91049

Summary: We propose a compound binomial model defined in a markovian environment which is an extension to the compound binomial model presented by H.-U. Gerber [Mathematical fun with the compound bionomial process. ASTIN Bull. 18, 109–123 (1998); Mathematical fun with ruin theory. Insur. Math. Econ. 7, 15–23 (1988)]. An algorithm is presented for the computation of the aggregate claim amount distribution for a fixed time period. We focus on infinite-time ruin probabilities and propose a numerical algorithm to compute their numerical values. Along the same lines as Gerber’s compound binomial model which can be used as an approximation to the classical risk model, we will see that the compound binomial model defined in a markovian environment can approximate the risk model based on a particular Cox model, the marked Markov modulated Poisson process. Finally, we compare via stochastic ordering theory our proposed model to two other risk models: Gerber’s compound binomial model and a mixed compound binomial model. Numerical examples are provided.

MSC:

91B30 Risk theory, insurance (MSC2010)
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References:

[1] Asmussen, S., Risk process in a markovian environment, Scand. Act. J., 66-100 (1989) · Zbl 0684.62073
[2] Brémaud, P., Point Processes and Queues: Martingale Dynamics (1981), Springer-Verlag · Zbl 0478.60004
[3] Cossette, H.; Landriault, D.; Marceau, E., Ruin probabilities in the compound Markov binomial risk model, Scand. Act. J., 301-323 (2003) · Zbl 1092.91040
[4] Cox, D. R., Some statistical methods connected with series of events, J. Roy. Stat. Soc. Ser., B17, 129-164 (1955) · Zbl 0067.37403
[5] Daley, D. J.; Vere-Jones, D., An Introduction to the Theory of Point Processes (1988), Springer-Verlag · Zbl 0657.60069
[6] Denuit, M.; Genest, C.; Marceau, E., Criteria for the stochastic ordering of random sums, with actuarial applications, Scand. Act. J., 3-17 (2002) · Zbl 1003.60022
[7] De Vylder, F.; Marceau, E., Classical numerical ruin probabilities, Scand. Act. J., 109-123 (1996) · Zbl 0880.62108
[8] Dickson, D. C.M.; Egidio Dos Reis, A. D.; Waters, H. R., Some stable algorithms in ruin theory and their applications, ASTIN Bull., 25, 153-175 (1995)
[9] Gerber, H. U., Mathematical fun with the compound binomial process, ASTIN Bull., 18, 161-168 (1988)
[10] Gerber, H. U., Mathematical fun with ruin theory, Ins. Math. Econ., 7, 15-23 (1988) · Zbl 0657.62121
[11] Grandell, J., Aspects of Risk Theory (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0717.62100
[12] Grandell, J., Mixed Poisson Processes (1997), Chapman & Hall: Chapman & Hall New York · Zbl 0922.60005
[13] Janssen, J.; Reinhard, J. M., Probabilités de ruine pour une classe de modèles de risque semi-markoviens, ASTIN Bull., 15, 123-133 (1985)
[14] Joe, H., Multivariate Models and Dependence Concepts (1997), Chapman & Hall: Chapman & Hall London · Zbl 0990.62517
[15] Kaas, R., van Heerwaarden, A.E., Goovaerts, M.J., 1994. Ordering of Actuarial Risks. CAIRE Education Series 1, Brussels; Kaas, R., van Heerwaarden, A.E., Goovaerts, M.J., 1994. Ordering of Actuarial Risks. CAIRE Education Series 1, Brussels
[16] Lehtonen, T.; Nyrhinen, H., On asymptotically efficient simulation of ruin probabilities in a markovian environment, Scand. Act. J., 60-75 (1992) · Zbl 0755.62080
[17] Lillo, R. E.; Semeraro, P., Stochastic bounds for discrete-time claim processes with correlated risks, Scand. Act. J., 1-13 (2004) · Zbl 1114.62111
[18] Meester, L. E.; Shanthikumar, J. G., Regularity of stochastic processes: a theory based on directional convexity, Prob. Eng. Info. Sci., 7, 343-360 (1993)
[19] Müller, A.; Pflug, G., Asymptotic ruin probabilities for risk processes with dependent increments, Ins. Math. Econ., 28, 381-392 (2001) · Zbl 1055.91055
[20] Nyrhinen, H., Rough descriptions of ruin for a general class of surplus processes, Adv. Appl. Prob., 1008-1026 (1998) · Zbl 0932.60046
[21] Reinhard, J. M., On a class of semi-Markov risk models obtained as classical risk models in a markovian environment, ASTIN Bull., 18, 161-168 (1984)
[22] Reinhard, J. M.; Snoussi, M., The severity of ruin in a discrete semi-Markov risk model, Stochastic Models, 18, 85-107 (2002) · Zbl 1011.91056
[23] Reinhard, J.M., Snoussi, M., 2004. A monotonically converging algorithm for the severity of ruin in a discrete semi-Markov risk model. Scand. Act. J., To appear.; Reinhard, J.M., Snoussi, M., 2004. A monotonically converging algorithm for the severity of ruin in a discrete semi-Markov risk model. Scand. Act. J., To appear. · Zbl 1101.60070
[24] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic Processes for Insurance and Finance (1999), Wiley: Wiley New York · Zbl 0940.60005
[25] Shaked, M.; Shanthikumar, J. G., Stochastic Orders and their Applications (1994), Academic Press: Academic Press Boston · Zbl 0806.62009
[26] Yuen, K. C.; Guo, J. Y., Ruin probabilities for time-correlated claims in the compound binomial model, Ins. Math. Econ., 29, 47-57 (2001) · Zbl 1074.91032
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