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 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 & 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.   Google Scholar

[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.  Google Scholar

[3]

S. AsmussenB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251.  doi: 10.1080/10920277.2009.10597549.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

Y. Chen and B. Bian, Optimal dividend policies for compound poisson process with self-exciting, working paper. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. U. GerberX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. WangB. 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.  Google Scholar

[30]

H. Zhu, Dynamic Programming and Variational Inequalities in Singular Stochastic Control, , Ph. D Thesis, Brown University, 1992.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

S. AsmussenB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251.  doi: 10.1080/10920277.2009.10597549.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

Y. Chen and B. Bian, Optimal dividend policies for compound poisson process with self-exciting, working paper. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. U. GerberX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[26]

H. Schmidli, Stochastic Control in Insurance, Springer, New York, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. WangB. 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.  Google Scholar

[30]

H. Zhu, Dynamic Programming and Variational Inequalities in Singular Stochastic Control, , Ph. D Thesis, Brown University, 1992.  Google Scholar

Figure 1.  A sample path of Hawkes process $ (N_t,\lambda_t) $ and the surplus process $ X_t $ without dividends
Figure 2.  Several optimal dividends payment strategy examples
Figure 3.  The value function
Figure 4.  The fitting barrier curve
Figure 5.  The value of $ V $ and $ V^c $ with $ \lambda = 0.5 $ and associated barrier points
Figure 6.  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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