The Expected Discounted Tax Payments on Dual Risk Model under a Dividend Threshold

Affiliation(s)

College of Sciences, Jiangxi Agricultural University, Nanchang, China.

School of Mathematics and Statistics, Nanjing Audit University, Nanjing, China.

College of Sciences, Jiangxi Agricultural University, Nanchang, China.

School of Mathematics and Statistics, Nanjing Audit University, Nanjing, China.

ABSTRACT

In this paper, we consider the dual risk model in which periodic taxation are paid according to a loss-carry-forward system and dividends are paid under a threshold strategy. We give an analytical approach to derive the expression of gδ(u) (i.e. the Laplace transform of the first upper exit time). We discuss the expected discounted tax payments for this model and obtain its corresponding integro-differential equations. Finally, for Erlang (2) inter-innovation distribution, closedform expressions for the expected discounted tax payments are given.

In this paper, we consider the dual risk model in which periodic taxation are paid according to a loss-carry-forward system and dividends are paid under a threshold strategy. We give an analytical approach to derive the expression of gδ(u) (i.e. the Laplace transform of the first upper exit time). We discuss the expected discounted tax payments for this model and obtain its corresponding integro-differential equations. Finally, for Erlang (2) inter-innovation distribution, closedform expressions for the expected discounted tax payments are given.

Cite this paper

Z. Liu, A. Zhang and C. Li, "The Expected Discounted Tax Payments on Dual Risk Model under a Dividend Threshold,"*Open Journal of Statistics*, Vol. 3 No. 2, 2013, pp. 136-144. doi: 10.4236/ojs.2013.32015.

Z. Liu, A. Zhang and C. Li, "The Expected Discounted Tax Payments on Dual Risk Model under a Dividend Threshold,"

References

[1] B. De Finetti, “Su un’Impostazione Alternativa Della Teoria Collettiva del Rischio,” Proceedings of the Transactions of the XV International Congress of Actuaries, New York, 1957, pp. 433-443.

[2] H. Bühlmann, “Mathematical Methods in Risk Theory,” Springer-Verlag, New York, Heidelberg. 1970.

[3] H. Gerber, “An Introduction to Mathematical Risk Theory. S. S. Huebner Foundation,” University of Pennsylvania, Philadelphia. 1979.

[4] H. Gerber and E. Shiu, “On the Time Value of Ruin,” North American Actuarial Journal, Vol. 2, No. 1, 1998, pp. 48-78. doi:10.1080/10920277.1998.10595671

[5] H. Gerber and E. Shiu, “On Optimal Dividend Strategies in the Compound Poisson Model,” Preprint, 2006.

[6] X. S. Lin, G. E. Willmot and S. Drekic, “The Classical Risk Model with a Constant Dividend Barrier: Analysis of the Gerber-Shiu Discounted Penalty Function,” Insurance: Mathematics and Economics, Vol. 33, No. 3, 2003, pp. 551-566. doi:10.1016/j.insmatheco.2003.08.004

[7] X. S. Lin and K. P. Pavlova, “The Compound Poisson Risk Model with a Threshold Dividend Strategy,” Insurance: Mathematics and Economics, Vol. 38, No. 1, 2005, pp. 57-80. doi:10.1016/j.insmatheco.2005.08.001

[8] D. C. M. Dickson and H. R. Waters, “Some Optimal Dividend Problems,” ASTIN Bulletin, Vol. 34 No. 1, 2004, pp. 49-74. doi:10.2143/AST.34.1.504954

[9] H. Albrecher, J. Hartinger and R. Tichy, “On the Distribution of Dividend Payments and the Discounted Penalty Function in a Risk Model with Linear Dividend Barrier,” Scandinavian Actuarial Journal, Vol. 2005, No. 2, 2005, pp. 103-126. doi:10.1080/03461230510006946

[10] Y. H. Dong, G. J. Wang and K. C. Yuen, “On the Renewal Risk Model under a Threshold Strategy,” Journal of Computational and Applied Mathematics, Vol. 230, No. 1, 2009, pp. 22-33.

[11] C. Y. Ng Andrew, “On a Dual Model with a Dividend Threshold,” Insurance: Mathematics and Economics, Vol. 44, No. 2, 2009, pp. 315-324. doi:10.1016/j.insmatheco.2008.11.011

[12] H. Albrecher, A. Badescu and D. Landriault, “On the Dual Risk Model with Taxation,” Insurance: Mathematics and Economics, Vol. 42, No. 3, 2008, pp. 1086-1094. doi:10.1016/j.insmatheco.2008.02.001

[13] H. Albrecher and C. Hipp, “Lundberg’s Risk Process with Tax,” Blätterder DGVFM, Vol. 28, No. 1, 2007, pp. 13-28. doi:10.1007/s11857-007-0004-4

[14] H. Albrecher, J. Renaud and X. W. Zhou, “A Lévy Insurance Risk Process with Tax,” Journal of Applied Probability, Vol. 45, No. 2, 2008, pp. 363-375. doi:10.1239/jap/1214950353

[15] 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

[16] W. Y. Wang and Y. J. Hu, “Optimal Loss-Carry-Forward Taxation for the Levy Risk Model,” Insurance: Mathematics and Economics, Vol. 50, No. 1, 2012, pp. 121-130. doi:10.1016/j.insmatheco.2011.10.011

[17] Z. Liu, W. Y. Wang and R. X. Ming, “The Threshold Dividend Strategy on a Class of Dual Model with Tax Payments,” Journal of University of Sience and Technology of China, 2013, in Press.

[18] Z. Liu, R. X. Ming, W. Y. Wang and X. Y. Song, “The Threshold Dividend Strategy in the Dual Risk Model Perturbed by Diffusion,” Journal of University of Sience and Technology of China, Vol. 42, No. 6, 2012, pp. 475-481.

[1] B. De Finetti, “Su un’Impostazione Alternativa Della Teoria Collettiva del Rischio,” Proceedings of the Transactions of the XV International Congress of Actuaries, New York, 1957, pp. 433-443.

[2] H. Bühlmann, “Mathematical Methods in Risk Theory,” Springer-Verlag, New York, Heidelberg. 1970.

[3] H. Gerber, “An Introduction to Mathematical Risk Theory. S. S. Huebner Foundation,” University of Pennsylvania, Philadelphia. 1979.

[4] H. Gerber and E. Shiu, “On the Time Value of Ruin,” North American Actuarial Journal, Vol. 2, No. 1, 1998, pp. 48-78. doi:10.1080/10920277.1998.10595671

[5] H. Gerber and E. Shiu, “On Optimal Dividend Strategies in the Compound Poisson Model,” Preprint, 2006.

[6] X. S. Lin, G. E. Willmot and S. Drekic, “The Classical Risk Model with a Constant Dividend Barrier: Analysis of the Gerber-Shiu Discounted Penalty Function,” Insurance: Mathematics and Economics, Vol. 33, No. 3, 2003, pp. 551-566. doi:10.1016/j.insmatheco.2003.08.004

[7] X. S. Lin and K. P. Pavlova, “The Compound Poisson Risk Model with a Threshold Dividend Strategy,” Insurance: Mathematics and Economics, Vol. 38, No. 1, 2005, pp. 57-80. doi:10.1016/j.insmatheco.2005.08.001

[8] D. C. M. Dickson and H. R. Waters, “Some Optimal Dividend Problems,” ASTIN Bulletin, Vol. 34 No. 1, 2004, pp. 49-74. doi:10.2143/AST.34.1.504954

[9] H. Albrecher, J. Hartinger and R. Tichy, “On the Distribution of Dividend Payments and the Discounted Penalty Function in a Risk Model with Linear Dividend Barrier,” Scandinavian Actuarial Journal, Vol. 2005, No. 2, 2005, pp. 103-126. doi:10.1080/03461230510006946

[10] Y. H. Dong, G. J. Wang and K. C. Yuen, “On the Renewal Risk Model under a Threshold Strategy,” Journal of Computational and Applied Mathematics, Vol. 230, No. 1, 2009, pp. 22-33.

[11] C. Y. Ng Andrew, “On a Dual Model with a Dividend Threshold,” Insurance: Mathematics and Economics, Vol. 44, No. 2, 2009, pp. 315-324. doi:10.1016/j.insmatheco.2008.11.011

[12] H. Albrecher, A. Badescu and D. Landriault, “On the Dual Risk Model with Taxation,” Insurance: Mathematics and Economics, Vol. 42, No. 3, 2008, pp. 1086-1094. doi:10.1016/j.insmatheco.2008.02.001

[13] H. Albrecher and C. Hipp, “Lundberg’s Risk Process with Tax,” Blätterder DGVFM, Vol. 28, No. 1, 2007, pp. 13-28. doi:10.1007/s11857-007-0004-4

[14] H. Albrecher, J. Renaud and X. W. Zhou, “A Lévy Insurance Risk Process with Tax,” Journal of Applied Probability, Vol. 45, No. 2, 2008, pp. 363-375. doi:10.1239/jap/1214950353

[15] 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

[16] W. Y. Wang and Y. J. Hu, “Optimal Loss-Carry-Forward Taxation for the Levy Risk Model,” Insurance: Mathematics and Economics, Vol. 50, No. 1, 2012, pp. 121-130. doi:10.1016/j.insmatheco.2011.10.011

[17] Z. Liu, W. Y. Wang and R. X. Ming, “The Threshold Dividend Strategy on a Class of Dual Model with Tax Payments,” Journal of University of Sience and Technology of China, 2013, in Press.

[18] Z. Liu, R. X. Ming, W. Y. Wang and X. Y. Song, “The Threshold Dividend Strategy in the Dual Risk Model Perturbed by Diffusion,” Journal of University of Sience and Technology of China, Vol. 42, No. 6, 2012, pp. 475-481.