Dividend Payments with a Hybrid Strategy in the Compound Poisson Risk Model
Affiliation(s)
School of Mathematical Sciences, Qufu Normal University, Qufu, China. China Institute for Actuarial Science, Central University of Finance and Economics, Beijing, China.
School of Mathematical Sciences, Qufu Normal University, Qufu, China. China Institute for Actuarial Science, Central University of Finance and Economics, Beijing, China.
ABSTRACT
In this paper, a hybrid dividend strategy in the compound Poisson risk model is considered. In the absence of dividends, the surplus of an insurance company is modelled by a compound Poisson process. Dividends are paid at a constant rate whenever the modified surplus is in a interval; the premium income no longer goes into the surplus but is paid out as dividends whenever the modified surplus exceeds the upper bound of the interval, otherwise no dividends are paid. Integro-differential equations with boundary conditions satisfied by the expected total discounted dividends until ruin are derived; for example, closed-form solutions are given when claims are exponentially distributed. Accordingly, the moments and moment-generating functions of total discounted dividends until ruin are considered. Finally, the Gerber-Shiu function and Laplace transform of the ruin time are discussed.
In this paper, a hybrid dividend strategy in the compound Poisson risk model is considered. In the absence of dividends, the surplus of an insurance company is modelled by a compound Poisson process. Dividends are paid at a constant rate whenever the modified surplus is in a interval; the premium income no longer goes into the surplus but is paid out as dividends whenever the modified surplus exceeds the upper bound of the interval, otherwise no dividends are paid. Integro-differential equations with boundary conditions satisfied by the expected total discounted dividends until ruin are derived; for example, closed-form solutions are given when claims are exponentially distributed. Accordingly, the moments and moment-generating functions of total discounted dividends until ruin are considered. Finally, the Gerber-Shiu function and Laplace transform of the ruin time are discussed.
KEYWORDS
Hybrid Dividend Strategy, Compound Poisson Risk Model, Moment-Generating Function, Gerber-Shiu Function
Hybrid Dividend Strategy, Compound Poisson Risk Model, Moment-Generating Function, Gerber-Shiu Function
Cite this paper
Li, P. , Yin, C. and Zhou, M. (2014) Dividend Payments with a Hybrid Strategy in the Compound Poisson Risk Model. Applied Mathematics, 5, 1933-1949. doi: 10.4236/am.2014.513187.
Li, P. , Yin, C. and Zhou, M. (2014) Dividend Payments with a Hybrid Strategy in the Compound Poisson Risk Model. Applied Mathematics, 5, 1933-1949. doi: 10.4236/am.2014.513187.
References
[1] Finetti, B.D. (1957) Su un impostazione alternativa della teoria collectiva del rischio. Transactions of the 15th International Congress of Applied Probability, 41, 117-130. [2] Gerber, H.U. and Shiu, E.S.W. (1998) On the Time Value of Ruin. North American Actuarial Journal, 2, 48-78.
http://dx.doi.org/10.1080/10920277.1998.10595671 [3] Lin, X.S., Willmot, G.E. and Drekic, S. (2003) The Classical Risk Model with a Constant Dividend Barrier: Analysis of the Gerber-Shiu Discounted Penalty Function. Insurance: Mathematics and Economics, 33, 551-566.
http://dx.doi.org/10.1016/j.insmatheco.2003.08.004 [4] Gerber, H.U. and Shiu, E.S.W. (2004) Optimal Dividends: Analysis with Brownian Motion. North American Actuarial Journal, 8, 1-20.
http://dx.doi.org/10.1080/10920277.2004.10596125 [5] Jeanblang-Picque, M. and Shiryaev, A.N. (1995) Optimization of the Flow of Dividends. Russian Mathematical Surveys, 20, 257-277.
http://dx.doi.org/10.1070/RM1995v050n02ABEH002054 [6] Asmussen, S. and Taksar, M. (1997) Controlled Diffusion Models for Optimal Dividend Pay-Out. Insurance: Mathematics and Economics, 20, 1-15.
http://dx.doi.org/10.1016/S0167-6687(96)00017-0 [7] Gerber, H.U. and Shiu, E.S.W. (2006) On Optimal Dividend Strategy in the Compound Poisson Model. North American Actuarial Journal, 10, 76-93.
http://dx.doi.org/10.1080/10920277.2006.10596249 [8] Gerber, H.U. and Shiu, E.S.W. (2006) On Optimal Dividends: From Reflection to Refraction. Journal of Computational and Applied Mathematics, 186, 4-22.
http://dx.doi.org/10.1016/j.cam.2005.03.062 [9] Lin, X.S. and Pavlova, K.P. (2006) The Compound Poisson Risk Model with a Threshold Dividend Strategy. Insurance: Mathematics and Economics, 38, 57-80.
http://dx.doi.org/10.1016/j.insmatheco.2005.08.001 [10] Wan, N. (2007) Dividend Payments with a Threshold Strategy in the Compound Poisson Risk Model Perturbed by Diffusion. Insurance: Mathematics and Economics, 40, 509-523.
http://dx.doi.org/10.1016/j.insmatheco.2006.08.002 [11] Ng, A.C.Y. (2009) On a Dual Model with a Dividend Threshold. Insurance: Mathematics and Economics, 44, 315-324.
http://dx.doi.org/10.1016/j.insmatheco.2008.11.011 [12] Fang, Y. and Wu, R. (2008) Optimal Dividends in the Brownian Motion Risk Model with Interest. Journal of Computational and Applied Mathematics, 229, 145-151.
http://dx.doi.org/10.1016/j.cam.2008.10.021 [13] Chi, Y.C. and Lin, X.S. (2011) On the Threshold Dividend Strategy for a Generalized Jump-Diffusion Risk Model. Insurance: Mathematics and Economics, 48, 326-337.
http://dx.doi.org/10.1016/j.insmatheco.2010.11.006 [14] Yin, C.C. and Wen, Y.Z. (2013) An Extension of Paulsen-Gjessing’s Risk Model with Stochastic Return on Investments. Insurance: Mathematics and Economics, 52, 469-476.
http://dx.doi.org/10.1016/j.insmatheco.2013.02.014 [15] Albrecher, H. and Hartinger, J. (2007) A Risk Model with Multi-Layer Dividend Strategy. North American Actuarial Journal, 11, 43-64.
http://dx.doi.org/10.1080/10920277.2007.10597447 [16] Lin, X.S. and Sendova, K.P. (2008) The Compound Poisson Risk Model with Multiple Threshold. Insurance: Mathematics and Economics, 42, 617-627.
http://dx.doi.org/10.1016/j.insmatheco.2007.06.008 [17] Jiang, W.Y., Yang, Z.J. and Li, X.P. (2012) The Discounted Penalty Function with Multi-Layer Dividend Strategy in the Phase-Type Risk Model. Insurance: Mathematics and Economics, 82, 1358-1366. [18] Ng, A.C.Y. (2010) On the Upcrossing and Downcrossing Probabilities of a Dual Risk Model with Phase-Type Gains. Astin Bulletin, 40, 281-306.
http://dx.doi.org/10.2143/AST.40.1.2049230 [19] Wang, C.W., Yin, C.C. and Li, E.Q. (2010) On the Classical Risk Model with Credit and Debit Interests under Absolute Ruin. Statistics and Probability Letter, 80, 427-436.
http://dx.doi.org/10.1016/j.spl.2009.11.020 [20] Liu, D.H. and Liu, Z.M. (2011) The Perturbed Compound Poisson Risk Model with Linear Dividend Barrier. Journal of Computational and Applied Mathematics, 235, 2357-2363.
http://dx.doi.org/10.1016/j.cam.2010.10.034 [21] Gao, S. and Liu, Z.M. (2010) The Perturbed Compound Poisson Risk Model with Constant Interest and a Threshold Dividend Strategy. Journal of Computational and Applied Mathematics, 233, 2181-2188.
http://dx.doi.org/10.1016/j.cam.2009.10.004
[1] Finetti, B.D. (1957) Su un impostazione alternativa della teoria collectiva del rischio. Transactions of the 15th International Congress of Applied Probability, 41, 117-130. [2] Gerber, H.U. and Shiu, E.S.W. (1998) On the Time Value of Ruin. North American Actuarial Journal, 2, 48-78.
http://dx.doi.org/10.1080/10920277.1998.10595671 [3] Lin, X.S., Willmot, G.E. and Drekic, S. (2003) The Classical Risk Model with a Constant Dividend Barrier: Analysis of the Gerber-Shiu Discounted Penalty Function. Insurance: Mathematics and Economics, 33, 551-566.
http://dx.doi.org/10.1016/j.insmatheco.2003.08.004 [4] Gerber, H.U. and Shiu, E.S.W. (2004) Optimal Dividends: Analysis with Brownian Motion. North American Actuarial Journal, 8, 1-20.
http://dx.doi.org/10.1080/10920277.2004.10596125 [5] Jeanblang-Picque, M. and Shiryaev, A.N. (1995) Optimization of the Flow of Dividends. Russian Mathematical Surveys, 20, 257-277.
http://dx.doi.org/10.1070/RM1995v050n02ABEH002054 [6] Asmussen, S. and Taksar, M. (1997) Controlled Diffusion Models for Optimal Dividend Pay-Out. Insurance: Mathematics and Economics, 20, 1-15.
http://dx.doi.org/10.1016/S0167-6687(96)00017-0 [7] Gerber, H.U. and Shiu, E.S.W. (2006) On Optimal Dividend Strategy in the Compound Poisson Model. North American Actuarial Journal, 10, 76-93.
http://dx.doi.org/10.1080/10920277.2006.10596249 [8] Gerber, H.U. and Shiu, E.S.W. (2006) On Optimal Dividends: From Reflection to Refraction. Journal of Computational and Applied Mathematics, 186, 4-22.
http://dx.doi.org/10.1016/j.cam.2005.03.062 [9] Lin, X.S. and Pavlova, K.P. (2006) The Compound Poisson Risk Model with a Threshold Dividend Strategy. Insurance: Mathematics and Economics, 38, 57-80.
http://dx.doi.org/10.1016/j.insmatheco.2005.08.001 [10] Wan, N. (2007) Dividend Payments with a Threshold Strategy in the Compound Poisson Risk Model Perturbed by Diffusion. Insurance: Mathematics and Economics, 40, 509-523.
http://dx.doi.org/10.1016/j.insmatheco.2006.08.002 [11] Ng, A.C.Y. (2009) On a Dual Model with a Dividend Threshold. Insurance: Mathematics and Economics, 44, 315-324.
http://dx.doi.org/10.1016/j.insmatheco.2008.11.011 [12] Fang, Y. and Wu, R. (2008) Optimal Dividends in the Brownian Motion Risk Model with Interest. Journal of Computational and Applied Mathematics, 229, 145-151.
http://dx.doi.org/10.1016/j.cam.2008.10.021 [13] Chi, Y.C. and Lin, X.S. (2011) On the Threshold Dividend Strategy for a Generalized Jump-Diffusion Risk Model. Insurance: Mathematics and Economics, 48, 326-337.
http://dx.doi.org/10.1016/j.insmatheco.2010.11.006 [14] Yin, C.C. and Wen, Y.Z. (2013) An Extension of Paulsen-Gjessing’s Risk Model with Stochastic Return on Investments. Insurance: Mathematics and Economics, 52, 469-476.
http://dx.doi.org/10.1016/j.insmatheco.2013.02.014 [15] Albrecher, H. and Hartinger, J. (2007) A Risk Model with Multi-Layer Dividend Strategy. North American Actuarial Journal, 11, 43-64.
http://dx.doi.org/10.1080/10920277.2007.10597447 [16] Lin, X.S. and Sendova, K.P. (2008) The Compound Poisson Risk Model with Multiple Threshold. Insurance: Mathematics and Economics, 42, 617-627.
http://dx.doi.org/10.1016/j.insmatheco.2007.06.008 [17] Jiang, W.Y., Yang, Z.J. and Li, X.P. (2012) The Discounted Penalty Function with Multi-Layer Dividend Strategy in the Phase-Type Risk Model. Insurance: Mathematics and Economics, 82, 1358-1366. [18] Ng, A.C.Y. (2010) On the Upcrossing and Downcrossing Probabilities of a Dual Risk Model with Phase-Type Gains. Astin Bulletin, 40, 281-306.
http://dx.doi.org/10.2143/AST.40.1.2049230 [19] Wang, C.W., Yin, C.C. and Li, E.Q. (2010) On the Classical Risk Model with Credit and Debit Interests under Absolute Ruin. Statistics and Probability Letter, 80, 427-436.
http://dx.doi.org/10.1016/j.spl.2009.11.020 [20] Liu, D.H. and Liu, Z.M. (2011) The Perturbed Compound Poisson Risk Model with Linear Dividend Barrier. Journal of Computational and Applied Mathematics, 235, 2357-2363.
http://dx.doi.org/10.1016/j.cam.2010.10.034 [21] Gao, S. and Liu, Z.M. (2010) The Perturbed Compound Poisson Risk Model with Constant Interest and a Threshold Dividend Strategy. Journal of Computational and Applied Mathematics, 233, 2181-2188.
http://dx.doi.org/10.1016/j.cam.2009.10.004