Optimal Dividend Problem for a Compound Poisson Risk Model
ABSTRACT
In this note we study the optimal dividend problem for a company whose surplus process, in the absence of dividend payments, evolves as a generalized compound Poisson model in which the counting process is a generalized Poisson process. This model includes the classical risk model and the Pólya-Aeppli risk model as special cases. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that under some conditions the optimal dividend strategy is formed by a barrier strategy. Moreover, two conjectures are proposed.
KEYWORDS
Barrier Strategy, Optimal Dividend Strategy, Generalized Compound Poisson Risk Model, Stochastic Control
Barrier Strategy, Optimal Dividend Strategy, Generalized Compound Poisson Risk Model, Stochastic Control
Cite this paper
Shen, Y. and Yin, C. (2014) Optimal Dividend Problem for a Compound Poisson Risk Model. Applied Mathematics, 5, 1496-1502. doi: 10.4236/am.2014.510142.
Shen, Y. and Yin, C. (2014) Optimal Dividend Problem for a Compound Poisson Risk Model. Applied Mathematics, 5, 1496-1502. doi: 10.4236/am.2014.510142.
References
[1] Finetti, B.D. (1957) Su un’impostazion alternativa dell teoria collecttiva del rischio. Transactions of the 15th International Congress of Actuaries, 2, 433-443. [2] Gerber, H.U. (1969) Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, 69, 185-227. [3] Azcue, P. and Muler, N. (2005) Optimal Reinsurance and Dividend Distribution Policies in the Cramér-Lundberg Model. Mathematical Finance, 15, 261-308.
http://dx.doi.org/10.1111/j.0960-1627.2005.00220.x [4] Albrecher, H. and Thonhauser, S. (2008) Optimal Dividend Strategies for a Risk Process under Force of Interest. Insurance: Mathematics and Economics, 43, 134-149.
http://dx.doi.org/10.1016/j.insmatheco.2008.03.012 [5] Avram, F., Palmowski, Z. and Pistorius, M.R. (2007) On the Optimal Dividend Problem for a Spectrally Negative Lévy Process. The Annals of Applied Probability, 17, 156-180.
http://dx.doi.org/10.1214/105051606000000709 [6] Loeffen, R. (2008) On Optimality of the Barrier Strategy in de Finetti’s Dividend Problem for Spectrally Negative Lévy Processes. The Annals of Applied Probability, 18, 1669-1680.
http://dx.doi.org/10.1214/07-AAP504 [7] Kyprianou, A.E., Rivero, V. and Song, R. (2010) Convexity and Smoothness of Scale Functions with Applications to de Finetti’s Control Problem. Journal of Theoretical Probability, 23, 547-564.
http://dx.doi.org/10.1007/s10959-009-0220-z [8] Yin, C.C. and Wang, C.W. (2009) Optimality of the Barrier Strategy in de Finetti’s Dividend Problem for Spectrally Negative Lévy Processes: An Alternative Approach. Journal of Computational and Applied Mathematics, 233, 482491.
http://dx.doi.org/10.1016/j.cam.2009.07.051 [9] Loeffen, R. and Renaud, J.F. (2010) De Finetti’s Optimal Dividends Problem with an Affine Penalty Function at Ruin. Insurance: Mathematics and Economics, 46, 98-108.
http://dx.doi.org/10.1016/j.insmatheco.2009.09.006 [10] Azcue, P. and Muler, N. (2010) Optimal Investment Policy and Dividend Payment Strategy in an Insurance Company. The Annals of Applied Probability, 20, 1253-1302.
http://dx.doi.org/10.1214/09-AAP643 [11] Chiu, S.N. and Yin, C.C. (2003) The Time of Ruin the Surplus Prior to Ruin and the Deficit at Ruin for the Classical Risk Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 33, 59-66.
http://dx.doi.org/10.1016/S0167-6687(03)00143-4 [12] Minkova, L.D. (2004) The Pólya-Aeppli Process and Ruin Problems. Journal of Applied Mathematics and Stochastic Analysis, 3, 221-234.
http://dx.doi.org/10.1155/S1048953304309032 [13] Mao, Z.C. and Liu, J.E. (2005) A Risk Model and Ruin Probability with Compound Poisson-Geometric Process. Acta Mathematicae Applicatae Sinica, 28, 419-428. (in Chinese) [14] Quenouille, M.H. (1949) A Relation between the Logarithmic, Poisson, and Negative Binomial Series. Biometrics, 5, 162-164.
http://dx.doi.org/10.2307/3001917 [15] Willmot, G.E. and Lin, X.S. (2001) Lundberg Approximations for Compound Distributions with Insurance Applications. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4613-0111-0 [16] Van Harn, K. (1978) Classifying Infinitely Divisible Distributions by Functional Equations. CWI, Amsterdam. [17] Hansen, B.G. and Willekens, E. (1990) The Generalized Logarithmic Series Distribution. Statistics & Probability Letters, 9, 311-316.
http://dx.doi.org/10.1016/0167-7152(90)90138-W [18] Chiu, S.N. and Yin, C.C. (2014) On the Complete Monotonicity of the Compound Geometric Convolution with Applications to Risk Theory. Scandinavian Actuarial Journal, 2014, 116-124.
http://dx.doi.org/10.1080/03461238.2011.647061 [19] Shanthikumar, J.G. (1988) DFR Property of First Passage Times and Its Preservation under Geometric Compounding. The Annals of Probability, 16, 397-406.
http://dx.doi.org/10.1214/aop/1176991910 [20] Esary, J.D. and Marshall, A.W. (1973) Shock Models and Wear Processes. The Annals of Probability, 1, 627-649.
http://dx.doi.org/10.1214/aop/1176996891
[1] Finetti, B.D. (1957) Su un’impostazion alternativa dell teoria collecttiva del rischio. Transactions of the 15th International Congress of Actuaries, 2, 433-443. [2] Gerber, H.U. (1969) Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, 69, 185-227. [3] Azcue, P. and Muler, N. (2005) Optimal Reinsurance and Dividend Distribution Policies in the Cramér-Lundberg Model. Mathematical Finance, 15, 261-308.
http://dx.doi.org/10.1111/j.0960-1627.2005.00220.x [4] Albrecher, H. and Thonhauser, S. (2008) Optimal Dividend Strategies for a Risk Process under Force of Interest. Insurance: Mathematics and Economics, 43, 134-149.
http://dx.doi.org/10.1016/j.insmatheco.2008.03.012 [5] Avram, F., Palmowski, Z. and Pistorius, M.R. (2007) On the Optimal Dividend Problem for a Spectrally Negative Lévy Process. The Annals of Applied Probability, 17, 156-180.
http://dx.doi.org/10.1214/105051606000000709 [6] Loeffen, R. (2008) On Optimality of the Barrier Strategy in de Finetti’s Dividend Problem for Spectrally Negative Lévy Processes. The Annals of Applied Probability, 18, 1669-1680.
http://dx.doi.org/10.1214/07-AAP504 [7] Kyprianou, A.E., Rivero, V. and Song, R. (2010) Convexity and Smoothness of Scale Functions with Applications to de Finetti’s Control Problem. Journal of Theoretical Probability, 23, 547-564.
http://dx.doi.org/10.1007/s10959-009-0220-z [8] Yin, C.C. and Wang, C.W. (2009) Optimality of the Barrier Strategy in de Finetti’s Dividend Problem for Spectrally Negative Lévy Processes: An Alternative Approach. Journal of Computational and Applied Mathematics, 233, 482491.
http://dx.doi.org/10.1016/j.cam.2009.07.051 [9] Loeffen, R. and Renaud, J.F. (2010) De Finetti’s Optimal Dividends Problem with an Affine Penalty Function at Ruin. Insurance: Mathematics and Economics, 46, 98-108.
http://dx.doi.org/10.1016/j.insmatheco.2009.09.006 [10] Azcue, P. and Muler, N. (2010) Optimal Investment Policy and Dividend Payment Strategy in an Insurance Company. The Annals of Applied Probability, 20, 1253-1302.
http://dx.doi.org/10.1214/09-AAP643 [11] Chiu, S.N. and Yin, C.C. (2003) The Time of Ruin the Surplus Prior to Ruin and the Deficit at Ruin for the Classical Risk Process Perturbed by Diffusion. Insurance: Mathematics and Economics, 33, 59-66.
http://dx.doi.org/10.1016/S0167-6687(03)00143-4 [12] Minkova, L.D. (2004) The Pólya-Aeppli Process and Ruin Problems. Journal of Applied Mathematics and Stochastic Analysis, 3, 221-234.
http://dx.doi.org/10.1155/S1048953304309032 [13] Mao, Z.C. and Liu, J.E. (2005) A Risk Model and Ruin Probability with Compound Poisson-Geometric Process. Acta Mathematicae Applicatae Sinica, 28, 419-428. (in Chinese) [14] Quenouille, M.H. (1949) A Relation between the Logarithmic, Poisson, and Negative Binomial Series. Biometrics, 5, 162-164.
http://dx.doi.org/10.2307/3001917 [15] Willmot, G.E. and Lin, X.S. (2001) Lundberg Approximations for Compound Distributions with Insurance Applications. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4613-0111-0 [16] Van Harn, K. (1978) Classifying Infinitely Divisible Distributions by Functional Equations. CWI, Amsterdam. [17] Hansen, B.G. and Willekens, E. (1990) The Generalized Logarithmic Series Distribution. Statistics & Probability Letters, 9, 311-316.
http://dx.doi.org/10.1016/0167-7152(90)90138-W [18] Chiu, S.N. and Yin, C.C. (2014) On the Complete Monotonicity of the Compound Geometric Convolution with Applications to Risk Theory. Scandinavian Actuarial Journal, 2014, 116-124.
http://dx.doi.org/10.1080/03461238.2011.647061 [19] Shanthikumar, J.G. (1988) DFR Property of First Passage Times and Its Preservation under Geometric Compounding. The Annals of Probability, 16, 397-406.
http://dx.doi.org/10.1214/aop/1176991910 [20] Esary, J.D. and Marshall, A.W. (1973) Shock Models and Wear Processes. The Annals of Probability, 1, 627-649.
http://dx.doi.org/10.1214/aop/1176996891