# American Institute of Mathematical Sciences

March  2021, 11(1): 1-22. doi: 10.3934/mcrf.2020024

## Optimal dividend policy in an insurance company with contagious arrivals of claims

 School of Mathematical Sciences, Tongji University, Shanghai 200092, China

Received  September 2019 Revised  December 2019 Published  March 2021 Early access  March 2020

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.

Citation: Yiling Chen, Baojun Bian. Optimal dividend policy in an insurance company with contagious arrivals of claims. Mathematical Control and Related Fields, 2021, 11 (1) : 1-22. doi: 10.3934/mcrf.2020024
##### 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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##### 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.
A sample path of Hawkes process $(N_t,\lambda_t)$ and the surplus process $X_t$ without dividends
Several optimal dividends payment strategy examples
The value function
The fitting barrier curve
The value of $V$ and $V^c$ with $\lambda = 0.5$ and associated barrier points
The barrier curve under different parameter settings (A) the decay rate $\alpha$ (B) the long-run average of the claim intensity $\bar\lambda$ (C) the premium rate $p$ (D) the constant discount factor $c$
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