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Optimal dividend policy in an insurance company with contagious arrivals of claims
School of Mathematical Sciences, Tongji University, Shanghai 200092, China |
In this paper we consider the optimal dividend problem for an insurance company whose surplus follows a classical Cramér-Lundberg process with a feature of self-exciting. A Hawkes process is applied so that the occurrence of a jump in the claims triggers more sequent jumps. We show that the optimal value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a given boundary condition and declare its concavity. We introduce a barrier curve strategy and verify its optimality. Finally, some numerical results are exhibited.
References:
[1] |
Y. Aït-Sahalia and T. R. Hurd,
Portfolio choice in markets with contagion, Journal of Financial Econometrics, 14 (2015), 1-28.
|
[2] |
P. Azcue and N. Muler,
Optimal investment policy and dividend payment strategy in an insurance company, The Annals of Applied Probability, 20 (2010), 1253-1302.
doi: 10.1214/09-AAP643. |
[3] |
S. Asmussen, B. Højgaard and M. Taksar,
Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.
doi: 10.1007/s007800050075. |
[4] |
P. Azcue and N. Muler,
Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Mathematical Finance, 15 (2005), 261-308.
doi: 10.1111/j.0960-1627.2005.00220.x. |
[5] |
H. Albrecher and S. Thonhauser,
Optimality results for dividend problems in insurance, RACSAM-Revista de la Real Academia de Ciencias Exactas, 103 (2009), 295-320.
doi: 10.1007/BF03191909. |
[6] |
B. Avanzi,
Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251.
doi: 10.1080/10920277.2009.10597549. |
[7] |
H. Albrecher P. Azcue and N. Muler,
Optimal dividend strategies for two collaborating insurance companies, Advances in Applied Probability, 49 (2017), 515-548.
doi: 10.1017/apr.2017.11. |
[8] |
O. Alvarez J. M. Lasry and P. L. Lions,
Convex viscosity solutions and state constraints, Journal de Mathématiques Pures et Appliquées, 76 (1997), 265-288.
doi: 10.1016/S0021-7824(97)89952-7. |
[9] |
M. G. Crandall and P. L. Lions,
Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[10] |
M. G. Crandall and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American mathematical society, 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[11] |
Y. Chen and B. Bian,
Optimal investment and dividend policy in an insurance company: A varied bound for dividend rates, Discrete & Continuous Dynamical Systems-Series B, 24 (2019), 5083-5105.
|
[12] |
Y. Chen and B. Bian, Optimal dividend policies for compound poisson process with self-exciting, working paper. |
[13] |
P. A. Forsyth and G. Labahn,
Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, Journal of Computational Finance, 11 (2007), 1-43.
|
[14] |
H. U. Gerber and E. S. W. Shiu,
On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93.
doi: 10.1080/10920277.2006.10596249. |
[15] |
X. Gao and L. Zhu,
Large deviations and applications for Markovian Hawkes processes with a large initial intensity, Bernoulli, 24 (2018), 2875-2905.
doi: 10.3150/17-BEJ948. |
[16] |
H. U. Gerber, X. S. Lin and H. Yang,
A note on the dividends-penalty identity and the optimal dividend barrier, ASTIN Bulletin: The Journal of the IAA, 36 (2006), 489-503.
doi: 10.1017/S0515036100014604. |
[17] |
A. G. Hawkes,
Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58 (1971), 83-90.
doi: 10.1093/biomet/58.1.83. |
[18] |
D. Hainaut,
Contagion modeling between the financial and insurance markets with time changed processes, Insurance: Mathematics and Economics, 74 (2017), 63-77.
doi: 10.1016/j.insmatheco.2017.02.011. |
[19] |
Z. Jiang and M. Pistorius,
Optimal dividend distribution under Markov regime switching, Finance and Stochastics, 16 (2012), 449-476.
doi: 10.1007/s00780-012-0174-3. |
[20] |
Z. Jiang,
Optimal dividend policy when cash reserves follow a jump-diffusion process under Markov-regime switching, Journal of Applied Probability, 52 (2015), 209-223.
doi: 10.1239/jap/1429282616. |
[21] |
N. Kulenko and H. Schmidli,
Optimal dividend strategies in a Cramér-Lundberg model with capital injections, Insurance: Mathematics and Economics, 43 (2008), 270-278.
doi: 10.1016/j.insmatheco.2008.05.013. |
[22] |
H. Meng and T. K. Siu,
Optimal mixed impulse-equity insurance control problem with reinsurance, SIAM Journal on Control and Optimization, 49 (2011), 254-279.
doi: 10.1137/090773167. |
[23] |
J. Paulsen,
Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs, SIAM Journal on Control and Optimization, 47 (2008), 2201-2226.
doi: 10.1137/070691632. |
[24] |
H. Pham,
Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, Journal of Mathematical Systems, Estimation and Control, 8 (1998), 1-27.
|
[25] |
G. Stabile and G. L. Torrisi,
Risk processes with non-stationary Hawkes claims arrivals, Methodology and Computing in Applied Probability, 12 (2010), 415-429.
doi: 10.1007/s11009-008-9110-6. |
[26] |
H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008. |
[27] |
H. Schmidli,
On capital injections and dividends with tax in a classical risk model, Insurance: Mathematics and Economics, 71 (2016), 138-144.
doi: 10.1016/j.insmatheco.2016.08.004. |
[28] |
S. Thonhauser and H. Albrecher,
Optimal dividend strategies for a compound Poisson process under transaction costs and power utility, Stochastic Models, 27 (2011), 120-140.
doi: 10.1080/15326349.2011.542734. |
[29] |
Y. Wang, B. Bian and J. Zhang,
Viscosity solutions of Integro-Differential equations and passport options in a Jump-Diffusion model, Journal of Optimization Theory and Applications, 161 (2014), 122-144.
doi: 10.1007/s10957-013-0382-9. |
[30] |
H. Zhu, Dynamic Programming and Variational Inequalities in Singular Stochastic Control, , Ph. D Thesis, Brown University, 1992. |
show all references
References:
[1] |
Y. Aït-Sahalia and T. R. Hurd,
Portfolio choice in markets with contagion, Journal of Financial Econometrics, 14 (2015), 1-28.
|
[2] |
P. Azcue and N. Muler,
Optimal investment policy and dividend payment strategy in an insurance company, The Annals of Applied Probability, 20 (2010), 1253-1302.
doi: 10.1214/09-AAP643. |
[3] |
S. Asmussen, B. Højgaard and M. Taksar,
Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.
doi: 10.1007/s007800050075. |
[4] |
P. Azcue and N. Muler,
Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Mathematical Finance, 15 (2005), 261-308.
doi: 10.1111/j.0960-1627.2005.00220.x. |
[5] |
H. Albrecher and S. Thonhauser,
Optimality results for dividend problems in insurance, RACSAM-Revista de la Real Academia de Ciencias Exactas, 103 (2009), 295-320.
doi: 10.1007/BF03191909. |
[6] |
B. Avanzi,
Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251.
doi: 10.1080/10920277.2009.10597549. |
[7] |
H. Albrecher P. Azcue and N. Muler,
Optimal dividend strategies for two collaborating insurance companies, Advances in Applied Probability, 49 (2017), 515-548.
doi: 10.1017/apr.2017.11. |
[8] |
O. Alvarez J. M. Lasry and P. L. Lions,
Convex viscosity solutions and state constraints, Journal de Mathématiques Pures et Appliquées, 76 (1997), 265-288.
doi: 10.1016/S0021-7824(97)89952-7. |
[9] |
M. G. Crandall and P. L. Lions,
Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society, 277 (1983), 1-42.
doi: 10.1090/S0002-9947-1983-0690039-8. |
[10] |
M. G. Crandall and P. L. Lions,
User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American mathematical society, 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[11] |
Y. Chen and B. Bian,
Optimal investment and dividend policy in an insurance company: A varied bound for dividend rates, Discrete & Continuous Dynamical Systems-Series B, 24 (2019), 5083-5105.
|
[12] |
Y. Chen and B. Bian, Optimal dividend policies for compound poisson process with self-exciting, working paper. |
[13] |
P. A. Forsyth and G. Labahn,
Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance, Journal of Computational Finance, 11 (2007), 1-43.
|
[14] |
H. U. Gerber and E. S. W. Shiu,
On optimal dividend strategies in the compound Poisson model, North American Actuarial Journal, 10 (2006), 76-93.
doi: 10.1080/10920277.2006.10596249. |
[15] |
X. Gao and L. Zhu,
Large deviations and applications for Markovian Hawkes processes with a large initial intensity, Bernoulli, 24 (2018), 2875-2905.
doi: 10.3150/17-BEJ948. |
[16] |
H. U. Gerber, X. S. Lin and H. Yang,
A note on the dividends-penalty identity and the optimal dividend barrier, ASTIN Bulletin: The Journal of the IAA, 36 (2006), 489-503.
doi: 10.1017/S0515036100014604. |
[17] |
A. G. Hawkes,
Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58 (1971), 83-90.
doi: 10.1093/biomet/58.1.83. |
[18] |
D. Hainaut,
Contagion modeling between the financial and insurance markets with time changed processes, Insurance: Mathematics and Economics, 74 (2017), 63-77.
doi: 10.1016/j.insmatheco.2017.02.011. |
[19] |
Z. Jiang and M. Pistorius,
Optimal dividend distribution under Markov regime switching, Finance and Stochastics, 16 (2012), 449-476.
doi: 10.1007/s00780-012-0174-3. |
[20] |
Z. Jiang,
Optimal dividend policy when cash reserves follow a jump-diffusion process under Markov-regime switching, Journal of Applied Probability, 52 (2015), 209-223.
doi: 10.1239/jap/1429282616. |
[21] |
N. Kulenko and H. Schmidli,
Optimal dividend strategies in a Cramér-Lundberg model with capital injections, Insurance: Mathematics and Economics, 43 (2008), 270-278.
doi: 10.1016/j.insmatheco.2008.05.013. |
[22] |
H. Meng and T. K. Siu,
Optimal mixed impulse-equity insurance control problem with reinsurance, SIAM Journal on Control and Optimization, 49 (2011), 254-279.
doi: 10.1137/090773167. |
[23] |
J. Paulsen,
Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs, SIAM Journal on Control and Optimization, 47 (2008), 2201-2226.
doi: 10.1137/070691632. |
[24] |
H. Pham,
Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, Journal of Mathematical Systems, Estimation and Control, 8 (1998), 1-27.
|
[25] |
G. Stabile and G. L. Torrisi,
Risk processes with non-stationary Hawkes claims arrivals, Methodology and Computing in Applied Probability, 12 (2010), 415-429.
doi: 10.1007/s11009-008-9110-6. |
[26] |
H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008. |
[27] |
H. Schmidli,
On capital injections and dividends with tax in a classical risk model, Insurance: Mathematics and Economics, 71 (2016), 138-144.
doi: 10.1016/j.insmatheco.2016.08.004. |
[28] |
S. Thonhauser and H. Albrecher,
Optimal dividend strategies for a compound Poisson process under transaction costs and power utility, Stochastic Models, 27 (2011), 120-140.
doi: 10.1080/15326349.2011.542734. |
[29] |
Y. Wang, B. Bian and J. Zhang,
Viscosity solutions of Integro-Differential equations and passport options in a Jump-Diffusion model, Journal of Optimization Theory and Applications, 161 (2014), 122-144.
doi: 10.1007/s10957-013-0382-9. |
[30] |
H. Zhu, Dynamic Programming and Variational Inequalities in Singular Stochastic Control, , Ph. D Thesis, Brown University, 1992. |





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