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

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  • 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.

    Mathematics Subject Classification: Primary 35J87, 91B30, 49J20; Secondary 49L25, 91B70, 49K20.


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  • 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] 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. 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.
    [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. 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.
    [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. 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.
    [30] H. Zhu, Dynamic Programming and Variational Inequalities in Singular Stochastic Control, , Ph. D Thesis, Brown University, 1992.
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