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 $
[1]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[2]

Bingyan Liu, Xiongbing Ye, Xianzhou Dong, Lei Ni. Branching improved Deep Q Networks for solving pursuit-evasion strategy solution of spacecraft. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021016

[3]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[4]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[5]

Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096

[6]

Yantao Wang, Linlin Su. Monotone and nonmonotone clines with partial panmixia across a geographical barrier. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4019-4037. doi: 10.3934/dcds.2020056

[7]

C. J. Price. A modified Nelder-Mead barrier method for constrained optimization. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020058

[8]

Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605

[9]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388

[10]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398

[11]

Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020179

[12]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368

[13]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002

[14]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[15]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[16]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[17]

Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021001

[18]

Shin-Ichiro Ei, Masayasu Mimura, Tomoyuki Miyaji. Reflection of a self-propelling rigid disk from a boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 803-817. doi: 10.3934/dcdss.2020229

[19]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115

[20]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (123)
  • HTML views (376)
  • Cited by (0)

Other articles
by authors

[Back to Top]