Abstract
In this paper, we study the compound binomial model in Markovian environment, which is proposed by Cossette, et al. (2003). We obtain the recursive formula of the joint distributions of T, X(T − 1) and |X(T)| (i.e., the time of ruin, the surplus before ruin and the deficit at ruin) by the method of mass function of up-crossing zero points, as given by Liu and Zhao (2007). By using the same method, the recursive formula of supremum distribution is obtained. An example is included to illustrate the results of the model.
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References
A Asmussen. Risk theory in a Markovian environment, Scand Actuar J, 1989, 69–100.
H Cossette, D Landriault, E Marceau. Ruin probabilities in the compound Markov binomial risk model, Scand Actuar J, 2003, 301–323.
H Cossette, D Landriault, E Marceau. Compound binomial risk model in a Markovian environment, Insurance Math Econom, 2004, 35: 425–443.
H U Gerber. Mathematical fun with compound binomial process, Astin Bull, 1988, 161–168.
H U Gerber, E S W Shiu. The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance Math Econom, 1997, 21: 129–137.
G X Liu, J Y Zhao. Joint distributions of some actuarial random vectors in the compound binomial model, Insurance Math Econom, 2007, 40: 95–103.
S Li, Y Lu. Moments of the dividend payments and related problems in a Markov-modulated risk model, North Amer Actuar J, 2007, 11(2): 65–76.
S Li, Y Lu. The decompositions of the discounted penalty functions and the dividends-penalty identity in a Markov-modulated risk model, Astin Bull, 2008, 38: 53–71.
S Li, Y Lu. The Markovian regime-switching risk model with a threshould dividend strategy, Insurance Math Econom, 2009, 44: 296–303.
Y Lu, Tsai, C L Cary. The expected discounted penalty at ruin for a Markov-modulated risk process by diffusion, North Amer Actuar J, 2007, 11(2): 136–152.
A C Y Ng, H L Yang. On the joint distribution of surplus before and after ruin under a Markovian regime swithing model, Stochastic Process Appl, 2006, 116: 244–266.
E S W Shiu. The probability of eventual ruin in a compound binomial model, Astin Bull, 1987, 19: 179–190.
L Wei, R Wu. The joint distribution of several important actuarial diagnostics in the classical risk model, Insurance Math Econom, 2002, 30: 451–462.
G E Willmot. Ruin probabilities in the compound binomial model, Insurance Math Econom, 1993, 12: 133–142.
R Wu, G J Wang, L Wei. Joint distributions of some actuarial random vectors containing the time of ruin, Insurance Math Econom, 2003, 33: 147–161.
H Yang, Z M Zhang, C M Lan. Ruin problems in a discrete Markov risk model, Statist Probab Lett, 2009, 79: 21–28.
K C Yuen, J Guo. Some results on the compound Markov binomial model, Scand Actuar J, 2006, 129–140.
J Zhu, H Yang. Ruin theory for a Markov regime-switching risk model under a threshould dividend strategy, Insurance Math Econom, 2008, 42: 311–318.
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Supported by the National Natural Science Foundation of China (10671176, 10771192, 70871103).
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Yu, Yb., Zhang, Lx. & Zhang, Y. Joint and supremum distributions in the compound binomial model with Markovian environment. Appl. Math. J. Chin. Univ. 26, 265–279 (2011). https://doi.org/10.1007/s11766-011-2473-4
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DOI: https://doi.org/10.1007/s11766-011-2473-4