Dividend Payments and Related Problems in a Markov-Dependent Insurance Risk Model under Absolute Ruin

ABSTRACT

In this paper, we study the dividend payments prior to absolute ruin in a Markov-dependent risk process in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain. A system of integro-differential equations with boundary conditions satisfied by the moment-generating function, the nth moment of the discounted dividend payments prior to absolute ruin and the discounted penalty function, given the initial environment state, are derived. In the two-state risk model, explicit solutions to the integro-differential equations satisfied by the nth moment of the discounted dividend payments prior to absolute ruin are obtained when the claim size distribution is exponentially distributed. Finally, the matrix form of systems of integro-differential equations satisfied by the discounted penalty function are presented.

In this paper, we study the dividend payments prior to absolute ruin in a Markov-dependent risk process in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain. A system of integro-differential equations with boundary conditions satisfied by the moment-generating function, the nth moment of the discounted dividend payments prior to absolute ruin and the discounted penalty function, given the initial environment state, are derived. In the two-state risk model, explicit solutions to the integro-differential equations satisfied by the nth moment of the discounted dividend payments prior to absolute ruin are obtained when the claim size distribution is exponentially distributed. Finally, the matrix form of systems of integro-differential equations satisfied by the discounted penalty function are presented.

KEYWORDS

Absolute Ruin, Markov-Dependent Insurance Risk Model, Debit Interest, Moment-Generating Function

Absolute Ruin, Markov-Dependent Insurance Risk Model, Debit Interest, Moment-Generating Function

Cite this paper

nullW. Yu and Y. Huang, "Dividend Payments and Related Problems in a Markov-Dependent Insurance Risk Model under Absolute Ruin,"*American Journal of Industrial and Business Management*, Vol. 1 No. 1, 2011, pp. 1-9. doi: 10.4236/ajibm.2011.11001.

nullW. Yu and Y. Huang, "Dividend Payments and Related Problems in a Markov-Dependent Insurance Risk Model under Absolute Ruin,"

References

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[2] S. Asmussen, “Risk Theory in a Markovian Environ- ment,” Scandinavian Actuarial Journal, No. 2, 1989, pp. 69- 100.

[3] A. Ng and H. Yang, “On the Joint Distribution of Surplus Prior and Immediately after Ruin under a Markovian Re- gime Switching Model,” Stochastic Processes and Their Applications, Vol. 116, No. 2, 2006, pp. 244-266. doi:10.1016/j.spa.2005.09.008

[4] S. M. Li and Y. Lu, “Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model,” North American Actuarial Journal, Vol. 11, No. 2, 2007, pp. 65-76.

[5] Y. Lu and S. Li, “The Markovian Regime-Switching Risk Model with a Threshold Dividend Strategy,” Insurance: Mathematics and Economics, Vol. 44, No. 2, 2009, pp. 296-303. doi:10.1016/j.insmatheco.2008.04.004

[6] J. Liu, J. C. Xu and H. C. Hu, “The Markov-Dependent Risk Model with a Threshold Dividend Strategy,” Wuhan University Journal of Natural Sciences, Vol. 16, No. 3, 2011, pp. 193-198. doi:10.1007/s11859-011-0736-9

[7] J. Zhu and H. Yang, “Ruin Theory for a Markov Regime-Switching Model under a Threshold Dividend Strategy,” Insurance: Mathematics and Economics, Vol. 42, No. 1, 2008, pp. 311-318. doi:10.1016/j.insmatheco.2007.03.004

[8] J. Q. Wei, H. L. Yang and R. M. Wang, “On the Markov- modulated Insurance Risk Model with Tax,” Blaetter der DGVFM, Vol. 31, No. 1, 2010, pp. 65-78. doi:10.1007/s11857-010-0104-4

[9] H. Albrecher and O. Boxma, “On the Discounted Penalty Function in a Markov-Dependent Risk Model,” Insurance Mathematics and Economics, Vol. 37, No. 2, 2005, pp. 650-672. doi:10.1016/j.insmatheco.2005.06.007

[10] J. Liu, J. C. Xu and Y. J. Hu, “On the Expected Discounted Penalty Function in a Markov-Dependent Risk Model with Constant Dividend Barrier,” Acta Mathematica Scientia, Vol. 30B, No. 5, 2010, pp. 1481-1491.

[11] J. Liu, J. C. Xu and H. C. Hu, “Dividend Payments with a Threshold Strategy in a Markov-Dependent Risk Model,” Wuhan University Journal of Natural Sciences, Vol. 16, No. 1, 2011, pp. 11-15. doi:10.1007/s11859-011-0703-5

[12] M. Zhou and C. Zhang, “Absolute Ruin under Classical Risk Model,” Acta Mathematicae Applicate Sinica, Vol. 28, No. 4, 2005, pp. 57-80.

[13] J. Cai, “On the Time Value of Absolute Ruin with Debit Interest,” Advances in Applied Probability, Vol. 39, No. 2, 2007, pp. 343-359. doi:10.1239/aap/1183667614

[14] H. U. Gerber and H. L. Yang, “Absolute Ruin Probabilities in a Jump Diffusion Risk Model with Investment,” North American Actuarial Journal, Vol. 11, No. 3, 2007, pp. 159-169.

[15] K. C. Yuen, M. Zhou and J. Y. Guo, “On a Risk Model with Debit Interest and Dividend Payments,” Statistics and Probability Letters, Vol. 78, No. 15, 2008, pp. 2426-2432. doi:10.1016/j.spl.2008.02.021

[16] H. L. Yuan and Y. J. Hu, “Absolute Ruin in the Compound Poisson Risk Model with Constant Dividend Barrier,” Statistics and Probability Letter, Vol. 78, No. 14, 2008, pp. 2086-2094. doi:10.1016/j.spl.2008.01.076

[17] C. W. Wang and C. C. Yin, “Dividend Payments in the Classical Risk Model under Absolute Ruin with Debit Interest,” Applied Stochastic Models in Business and Industry, Vol. 25, No. 3, 2009, pp. 247-262. doi:10.1002/ asmb.722

[18] R. X. Ming, W. Y. Wang and L. Q. Xiao, “On the Time Value of Absolute Ruin with Tax,” Insurance: Mathematics and Economics, Vol. 46, No. 1, 2010, pp. 67-84. doi:10.1016/j.insmatheco.2009.09.004

[19] C. W. Wang, C. C. Yin and E. Q. Li, “On the Classical Risk Model with Credit and Debit Interests under Abso- lute Ruin,” Statistics and Probability Letters, Vol. 80, No. 15, 2010, pp. 427-436. doi:10.1016/j.spl.2009.11.020

[20] Z. M. Zhang, H. L. Yang and H. Yang, “On the Absolute Ruin in a Map Risk Model with Debit Interest,” Advances in Applied Probability, Vol. 43, No. 1, 2011, pp. 77-96. doi:10.1239/aap/1300198513

[21] W. G. Yu and Y. J. Huang, “Absolute Ruin for a Risk Model with Credit and Debit Interest under a Threshold Dividend Strategy,” Far East Journal of Applied Mathematics, Vol. 57, No. 2, 2011, pp. 125-137.

[1] J. M. Reinhard, “On a Class of Semi-Markov Risk Models Obtained as Classical Risk Models in a Markovian Enviroment,” Astin Bulletin, Vol. 14, 1984, pp. 23-43.

[2] S. Asmussen, “Risk Theory in a Markovian Environ- ment,” Scandinavian Actuarial Journal, No. 2, 1989, pp. 69- 100.

[3] A. Ng and H. Yang, “On the Joint Distribution of Surplus Prior and Immediately after Ruin under a Markovian Re- gime Switching Model,” Stochastic Processes and Their Applications, Vol. 116, No. 2, 2006, pp. 244-266. doi:10.1016/j.spa.2005.09.008

[4] S. M. Li and Y. Lu, “Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model,” North American Actuarial Journal, Vol. 11, No. 2, 2007, pp. 65-76.

[5] Y. Lu and S. Li, “The Markovian Regime-Switching Risk Model with a Threshold Dividend Strategy,” Insurance: Mathematics and Economics, Vol. 44, No. 2, 2009, pp. 296-303. doi:10.1016/j.insmatheco.2008.04.004

[6] J. Liu, J. C. Xu and H. C. Hu, “The Markov-Dependent Risk Model with a Threshold Dividend Strategy,” Wuhan University Journal of Natural Sciences, Vol. 16, No. 3, 2011, pp. 193-198. doi:10.1007/s11859-011-0736-9

[7] J. Zhu and H. Yang, “Ruin Theory for a Markov Regime-Switching Model under a Threshold Dividend Strategy,” Insurance: Mathematics and Economics, Vol. 42, No. 1, 2008, pp. 311-318. doi:10.1016/j.insmatheco.2007.03.004

[8] J. Q. Wei, H. L. Yang and R. M. Wang, “On the Markov- modulated Insurance Risk Model with Tax,” Blaetter der DGVFM, Vol. 31, No. 1, 2010, pp. 65-78. doi:10.1007/s11857-010-0104-4

[9] H. Albrecher and O. Boxma, “On the Discounted Penalty Function in a Markov-Dependent Risk Model,” Insurance Mathematics and Economics, Vol. 37, No. 2, 2005, pp. 650-672. doi:10.1016/j.insmatheco.2005.06.007

[10] J. Liu, J. C. Xu and Y. J. Hu, “On the Expected Discounted Penalty Function in a Markov-Dependent Risk Model with Constant Dividend Barrier,” Acta Mathematica Scientia, Vol. 30B, No. 5, 2010, pp. 1481-1491.

[11] J. Liu, J. C. Xu and H. C. Hu, “Dividend Payments with a Threshold Strategy in a Markov-Dependent Risk Model,” Wuhan University Journal of Natural Sciences, Vol. 16, No. 1, 2011, pp. 11-15. doi:10.1007/s11859-011-0703-5

[12] M. Zhou and C. Zhang, “Absolute Ruin under Classical Risk Model,” Acta Mathematicae Applicate Sinica, Vol. 28, No. 4, 2005, pp. 57-80.

[13] J. Cai, “On the Time Value of Absolute Ruin with Debit Interest,” Advances in Applied Probability, Vol. 39, No. 2, 2007, pp. 343-359. doi:10.1239/aap/1183667614

[14] H. U. Gerber and H. L. Yang, “Absolute Ruin Probabilities in a Jump Diffusion Risk Model with Investment,” North American Actuarial Journal, Vol. 11, No. 3, 2007, pp. 159-169.

[15] K. C. Yuen, M. Zhou and J. Y. Guo, “On a Risk Model with Debit Interest and Dividend Payments,” Statistics and Probability Letters, Vol. 78, No. 15, 2008, pp. 2426-2432. doi:10.1016/j.spl.2008.02.021

[16] H. L. Yuan and Y. J. Hu, “Absolute Ruin in the Compound Poisson Risk Model with Constant Dividend Barrier,” Statistics and Probability Letter, Vol. 78, No. 14, 2008, pp. 2086-2094. doi:10.1016/j.spl.2008.01.076

[17] C. W. Wang and C. C. Yin, “Dividend Payments in the Classical Risk Model under Absolute Ruin with Debit Interest,” Applied Stochastic Models in Business and Industry, Vol. 25, No. 3, 2009, pp. 247-262. doi:10.1002/ asmb.722

[18] R. X. Ming, W. Y. Wang and L. Q. Xiao, “On the Time Value of Absolute Ruin with Tax,” Insurance: Mathematics and Economics, Vol. 46, No. 1, 2010, pp. 67-84. doi:10.1016/j.insmatheco.2009.09.004

[19] C. W. Wang, C. C. Yin and E. Q. Li, “On the Classical Risk Model with Credit and Debit Interests under Abso- lute Ruin,” Statistics and Probability Letters, Vol. 80, No. 15, 2010, pp. 427-436. doi:10.1016/j.spl.2009.11.020

[20] Z. M. Zhang, H. L. Yang and H. Yang, “On the Absolute Ruin in a Map Risk Model with Debit Interest,” Advances in Applied Probability, Vol. 43, No. 1, 2011, pp. 77-96. doi:10.1239/aap/1300198513

[21] W. G. Yu and Y. J. Huang, “Absolute Ruin for a Risk Model with Credit and Debit Interest under a Threshold Dividend Strategy,” Far East Journal of Applied Mathematics, Vol. 57, No. 2, 2011, pp. 125-137.