# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018154

## Optimal investment and dividend for an insurer under a Markov regime switching market with high gain tax

 1 School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China 2 School of Finance, Nanjing University of Finance and Economics, Nanjing 210023, China 3 School of Economics, Nanjing University of Finance and Economics, Nanjing 210023, China

Received  March 2018 Revised  May 2018 Published  September 2018

This study examines the optimal investment and dividend problem for an insurer with CRRA preference. The insurer's goal is to maximize the expected discounted accumulated utility from dividend before ruin and the insurer subjects to high gain tax payment. Both the surplus process and the financial market are modulated by an external Markov chain. Using the weak dynamic programming principle (WDPP), we prove that the value function of our control problem is the unique viscosity solution to coupled Hamilton-Jacobi-Bellman (HJB) equations with first derivative constraints. Solving an auxiliary problem without regime switching, we prove that, it is optimal for the insurer in a multiple-regime market to adopt the policies in the same way as in a single-regime market. The regularity of the viscosity solution on its domain is proved and thus the HJB equations admits classical solution. A numerical scheme for the value function is provided by the Markov chain approximation method, two numerical examples are given to illustrate the impact of the high gain tax and regime switching on the optimal policies.

Citation: Lin Xu, Dingjun Yao, Gongpin Cheng. Optimal investment and dividend for an insurer under a Markov regime switching market with high gain tax. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018154
##### References:
 [1] B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251. doi: 10.1080/10920277.2009.10597549. [2] F. Avram, Z. Palmowski and M. Pistorius, On gerber-shiu functions and optimal dividend distribution for a lévy risk process in the presence of a penalty function, The Annals of Applied Probability, 25 (2015), 1868-1935. doi: 10.1214/14-AAP1038. [3] 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. [4] P. Azcue and N. Muler, Optimal dividend policies for compound poisson processes: The case of bounded dividend rates, Insurance: Mathematics and Economics, 51 (2012), 26-42. doi: 10.1016/j.insmatheco.2012.02.011. [5] L. Bai and J. Guo, Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes, Scandinavian Actuarial Journal, 2010 (2010), 36-55. doi: 10.1080/03461230802591098. [6] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited, Annales de l'IHP Analyse non linéaire, 25 (2008), 567-585. doi: 10.1016/j.anihpc.2007.02.007. [7] A. Barth, S. M. Bromberg and O. Reichmann, A non-stationary model of dividend distribution in a stochastic interest-rate setting, Computational Economics, 47 (2016), 447-472. [8] N. Bäuerle and U. Rieder, Portfolio optimization with markov-modulated stock prices and interest rates, IEEE Transactions on Automatic Control, 49 (2004), 442-447. doi: 10.1109/TAC.2004.824471. [9] B. Hamadéne, R. Belfadli, S. Hamadéne and Y. Ouknine, On one-dimensional stochastic differential equations involving the maximum process, Stochastics and Dynamics, 9 (2009), 277-292. doi: 10.1142/S0219493709002671. [10] B. Bouchard and N. Touzi, Weak dynamic programming principle for viscosity solutions, SIAM Journal on Control and Optimization, 49 (2011), 948-962. doi: 10.1137/090752328. [11] J. Brinkhuis and V. Tikhomirov, Optimization: Insights and Applications, Princeton University Press, 2005. doi: 10.1515/9781400829361. [12] A. Cadenillas, S. Sarkar and F. Zapatero, Optimal dividend policy with mean-reverting cash reservoir, Mathematical Finance, 17 (2007), 81-109. doi: 10.1111/j.1467-9965.2007.00295.x. [13] J. Cai, H. U. Gerber and H. Yang, Optimal dividends in an ornstein-uhlenbeck type model with credit and debit interest, North American Actuarial Journal, 10 (2006), 94-119. doi: 10.1080/10920277.2006.10596250. [14] S. Chen, Z. Li and Y. Zeng, Optimal dividend strategies with time-inconsistent preferences, Journal of Economic Dynamics and Control, 46 (2014), 150-172. doi: 10.1016/j.jedc.2014.06.018. [15] T. Choulli, M. Taksar and X. Zhou, A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979. doi: 10.1137/S0363012900382667. [16] K. L. Chung, A Course in Probability Theory, Academic press, 2001. [17] H. M. Clements and M. Krolzig, Can regime-switching models reproduce the business cycle features of us aggregate consumption, investment and output?, International Journal of Finance and Economics, 9 (2004), 1-14. doi: 10.1002/ijfe.231. [18] M. Crandall, H. Ishii 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. [19] Finetti De, Su un'impostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [20] D. Duffie, W. Fleming, H. M. Soner and T. Zariphopoulou, Hedging in incomplete markets with hara utility, Journal of Economic Dynamics and Control, 21 (1997), 753-782. doi: 10.1016/S0165-1889(97)00002-X. [21] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models, Springer, 1995. [22] R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer, 2005. [23] W. H. Fleming and T. Pang, A stochastic control model of investment, production and consumption, Quarterly of Applied Mathematics, 63 (2005), 71-87. doi: 10.1090/S0033-569X-04-00941-1. [24] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 2006. [25] J. Fu, J. Wei and H. Yang, Portfolio optimization in a regime-switching market with derivatives, European Journal of Operational Research, 233 (2014), 184-192. doi: 10.1016/j.ejor.2013.08.033. [26] J. Gaier, P. Grandits and W. Schachermayer, Asymptotic ruin probabilities and optimal investment, Annals of Applied Probability, 13 (2003), 1054-1076. doi: 10.1214/aoap/1060202834. [27] H. U. Gerber and E. S. W. Shiu, Optimal dividends: analysis with brownian motion, North American Actuarial Journal, 8 (2004), 1-20. doi: 10.1080/10920277.2004.10596125. [28] J. Grandell, Aspects of Risk Theory, Springer, 1991. doi: 10.1007/978-1-4613-9058-9. [29] P. Grandits, F. Hubalek, W. Schachermayer and M. Žigo, Optimal expected exponential utility of dividend payments in a brownian risk model, Scandinavian Actuarial Journal, 2007 (2007), 73-107. doi: 10.1080/03461230601165201. [30] B. Jgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: insurance company example, Mathematical Finance, 9 (1999), 153-182. doi: 10.1111/1467-9965.00066. [31] B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 4 (2004), 315-327. doi: 10.1088/1469-7688/4/3/007. [32] F. Hubalek and W. Schachermayer, Optimizing expected utility of dividend payments for a brownian risk process and a peculiar nonlinear model, Insurance: Mathematics and Economics, 34 (2004), 193-225. doi: 10.1016/j.insmatheco.2003.12.001. [33] K. Janecek and M. Sîrbu, Optimal investment with high-watermark performance fee, SIAM Journal on Control and Optimization, 50 (2012), 790-819. doi: 10.1137/100790884. [34] B. Jang and K. Kim, Optimal reinsurance and asset allocation under regime switching, Journal of Banking and Finance, 56 (2015), 37-47. doi: 10.1016/j.jbankfin.2015.03.002. [35] 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. [36] Z. Jin and G. Yin, Numerical methods for optimal dividend payment and investment strategies of Markov-modulated jump diffusion models with regular and singular controls, Journal of Optimization Theory and Applications, 159 (2013), 246-271. doi: 10.1007/s10957-012-0263-7. [37] Z. Jin, H. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica J. IFAC, 49 (2013), 2317-2329. doi: 10.1016/j.automatica.2013.04.043. [38] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Springer, 1987. [39] H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, 2001. doi: 10.1007/978-1-4613-0007-6. [40] G. Leobacher, M. Szölgyenyi and S. Thonhauser, Bayesian dividend optimization and finite time ruin probabilities, Stochastic Models, 30 (2014), 216-249. doi: 10.1080/15326349.2014.900390. [41] R. C Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. [42] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer Science and Business Media, 2013. [43] J. Paulsen, Optimal dividend payouts for diffusions with solvency constraints, Finance and Stochastics, 7 (2003), 457-473. doi: 10.1007/s007800200098. [44] I. Pospelov and S. Radionov, Optimal dividend policy when cash surplus follows telegraph process, 2015. [45] P. E. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, Berlin, 2005. doi: 10.1007/978-3-662-10061-5. [46] M. Reppen, J. Rochet and H. M. Soner. Optimal dividend policies with random profitability, arXiv preprint, arXiv: 1706.01813, 2017. [47] L. C. G. Rogers, Optimal Investment, Springer, 2013. doi: 10.1007/978-3-642-35202-7. [48] T. Rolski, H. Schmidli, V. Schmidt and J. Teugels, Stochastic Processes for Insurance and Finance, John Wiley & Sons, Ltd., Chichester, 1999. doi: 10.1002/9780470317044. [49] J. Sass and U. G. Haussmann, Optimizing the terminal wealth under partial information: The drift process as a continuous time markov chain, Finance and Stochastics, 8 (2004), 553-577. doi: 10.1007/s00780-004-0132-9. [50] Q. Song and C. Zhu, On singular control problems with state constraints and regime-switching: a viscosity solution approach, Automatica, 70 (2016), 66-73. doi: 10.1016/j.automatica.2016.03.017. [51] Q. Song, G. Yin and Z. Zhang, Numerical methods for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42 (2006), 1147-1157. doi: 10.1016/j.automatica.2006.03.016. [52] J. Stiglitz, Some aspects of the taxation of capital gains, Journal of Public Economics, 21 (1983), 257-294. doi: 10.3386/w1094. [53] M. Szölgyenyi, Dividend maximization in a hidden markov switching model, Statistics and Risk Modeling, 32 (2015), 143-158. doi: 10.1515/strm-2015-0019. [54] M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42. doi: 10.1007/s001860050001. [55] S. Tan, Z. Jin and G. Yin, Optimal dividend payment strategies with debt constraint in a hybrid regime-switching jump-diffusion model, Nonlinear Analysis: Hybrid Systems, 27 (2018), 141-156. doi: 10.1016/j.nahs.2017.08.007. [56] A. D. Wentzell, S. Chomet and K. L. Chung, A Course in the Theory of Stochastic Processes, McGraw-Hill International New York, 1981. [57] G. Xu and S. Shreve, A duality method for optimal consumption and investment under short-selling prohibition and general market coefficients, The Annals of Applied Probability, 2 (1992), 87-112. doi: 10.1214/aoap/1177005772. [58] D. Yao, H. Yang and R. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European Journal of Operational Research, 211 (2011), 568-576. doi: 10.1016/j.ejor.2011.01.015. [59] C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773. doi: 10.1016/j.insmatheco.2013.09.019. [60] K. C. Yiu, J. Liu, T. K. Siu and W. K. Ching, Optimal portfolios with regime switching and value-at-risk constraint, Automatica, 46 (2010), 979-989. doi: 10.1016/j.automatica.2010.02.027. [61] X. Zhang and T. K. Siu, On optimal proportional reinsurance and investment in a markovian regime-switching economy, Acta Mathematica Sinica, English Series, 28 (2012), 67-82. doi: 10.1007/s10114-012-9761-7. [62] Y. Zhao, R. Wang, D. Yao and P. Chen, Optimal dividends and capital injections in the dual model with a random time horizon, Journal of Optimization Theory and Applications, 167 (2015), 272-295. doi: 10.1007/s10957-014-0653-0. [63] L. Zheng, Optimal investment with high-watermark fee in a multi-dimensional jump diffusion model(doctoral dissertation), 2017. [64] J. Zhu, Singular optimal dividend control for the regime-switching Cramér-Lundberg model with credit and debit interest, Journal of Computational and Applied Mathematics, 257 (2014), 212-239. doi: 10.1016/j.cam.2013.08.033. [65] J. Zhu and F. Chen, Dividend optimization for regime-switching general diffusions, Insurance: Mathematics and Economics, 53 (2013), 439-456. doi: 10.1016/j.insmatheco.2013.07.006. [66] B. Zou and A. Cadenillas, Explicit solutions of optimal consumption, investment and insurance problems with regime switching, Insurance: Mathematics and Economics, 58 (2014), 159-167. doi: 10.1016/j.insmatheco.2014.07.006. [67] B. Zou and A. Cadenillas, Optimal investment and liability ratio policies in a multidimensional regime switching model, Risks, 5 (2017), p6. doi: 10.3390/risks5010006.

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##### References:
 [1] B. Avanzi, Strategies for dividend distribution: A review, North American Actuarial Journal, 13 (2009), 217-251. doi: 10.1080/10920277.2009.10597549. [2] F. Avram, Z. Palmowski and M. Pistorius, On gerber-shiu functions and optimal dividend distribution for a lévy risk process in the presence of a penalty function, The Annals of Applied Probability, 25 (2015), 1868-1935. doi: 10.1214/14-AAP1038. [3] 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. [4] P. Azcue and N. Muler, Optimal dividend policies for compound poisson processes: The case of bounded dividend rates, Insurance: Mathematics and Economics, 51 (2012), 26-42. doi: 10.1016/j.insmatheco.2012.02.011. [5] L. Bai and J. Guo, Optimal dividend payments in the classical risk model when payments are subject to both transaction costs and taxes, Scandinavian Actuarial Journal, 2010 (2010), 36-55. doi: 10.1080/03461230802591098. [6] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited, Annales de l'IHP Analyse non linéaire, 25 (2008), 567-585. doi: 10.1016/j.anihpc.2007.02.007. [7] A. Barth, S. M. Bromberg and O. Reichmann, A non-stationary model of dividend distribution in a stochastic interest-rate setting, Computational Economics, 47 (2016), 447-472. [8] N. Bäuerle and U. Rieder, Portfolio optimization with markov-modulated stock prices and interest rates, IEEE Transactions on Automatic Control, 49 (2004), 442-447. doi: 10.1109/TAC.2004.824471. [9] B. Hamadéne, R. Belfadli, S. Hamadéne and Y. Ouknine, On one-dimensional stochastic differential equations involving the maximum process, Stochastics and Dynamics, 9 (2009), 277-292. doi: 10.1142/S0219493709002671. [10] B. Bouchard and N. Touzi, Weak dynamic programming principle for viscosity solutions, SIAM Journal on Control and Optimization, 49 (2011), 948-962. doi: 10.1137/090752328. [11] J. Brinkhuis and V. Tikhomirov, Optimization: Insights and Applications, Princeton University Press, 2005. doi: 10.1515/9781400829361. [12] A. Cadenillas, S. Sarkar and F. Zapatero, Optimal dividend policy with mean-reverting cash reservoir, Mathematical Finance, 17 (2007), 81-109. doi: 10.1111/j.1467-9965.2007.00295.x. [13] J. Cai, H. U. Gerber and H. Yang, Optimal dividends in an ornstein-uhlenbeck type model with credit and debit interest, North American Actuarial Journal, 10 (2006), 94-119. doi: 10.1080/10920277.2006.10596250. [14] S. Chen, Z. Li and Y. Zeng, Optimal dividend strategies with time-inconsistent preferences, Journal of Economic Dynamics and Control, 46 (2014), 150-172. doi: 10.1016/j.jedc.2014.06.018. [15] T. Choulli, M. Taksar and X. Zhou, A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979. doi: 10.1137/S0363012900382667. [16] K. L. Chung, A Course in Probability Theory, Academic press, 2001. [17] H. M. Clements and M. Krolzig, Can regime-switching models reproduce the business cycle features of us aggregate consumption, investment and output?, International Journal of Finance and Economics, 9 (2004), 1-14. doi: 10.1002/ijfe.231. [18] M. Crandall, H. Ishii 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. [19] Finetti De, Su un'impostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. [20] D. Duffie, W. Fleming, H. M. Soner and T. Zariphopoulou, Hedging in incomplete markets with hara utility, Journal of Economic Dynamics and Control, 21 (1997), 753-782. doi: 10.1016/S0165-1889(97)00002-X. [21] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models, Springer, 1995. [22] R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer, 2005. [23] W. H. Fleming and T. Pang, A stochastic control model of investment, production and consumption, Quarterly of Applied Mathematics, 63 (2005), 71-87. doi: 10.1090/S0033-569X-04-00941-1. [24] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 2006. [25] J. Fu, J. Wei and H. Yang, Portfolio optimization in a regime-switching market with derivatives, European Journal of Operational Research, 233 (2014), 184-192. doi: 10.1016/j.ejor.2013.08.033. [26] J. Gaier, P. Grandits and W. Schachermayer, Asymptotic ruin probabilities and optimal investment, Annals of Applied Probability, 13 (2003), 1054-1076. doi: 10.1214/aoap/1060202834. [27] H. U. Gerber and E. S. W. Shiu, Optimal dividends: analysis with brownian motion, North American Actuarial Journal, 8 (2004), 1-20. doi: 10.1080/10920277.2004.10596125. [28] J. Grandell, Aspects of Risk Theory, Springer, 1991. doi: 10.1007/978-1-4613-9058-9. [29] P. Grandits, F. Hubalek, W. Schachermayer and M. Žigo, Optimal expected exponential utility of dividend payments in a brownian risk model, Scandinavian Actuarial Journal, 2007 (2007), 73-107. doi: 10.1080/03461230601165201. [30] B. Jgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: insurance company example, Mathematical Finance, 9 (1999), 153-182. doi: 10.1111/1467-9965.00066. [31] B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 4 (2004), 315-327. doi: 10.1088/1469-7688/4/3/007. [32] F. Hubalek and W. Schachermayer, Optimizing expected utility of dividend payments for a brownian risk process and a peculiar nonlinear model, Insurance: Mathematics and Economics, 34 (2004), 193-225. doi: 10.1016/j.insmatheco.2003.12.001. [33] K. Janecek and M. Sîrbu, Optimal investment with high-watermark performance fee, SIAM Journal on Control and Optimization, 50 (2012), 790-819. doi: 10.1137/100790884. [34] B. Jang and K. Kim, Optimal reinsurance and asset allocation under regime switching, Journal of Banking and Finance, 56 (2015), 37-47. doi: 10.1016/j.jbankfin.2015.03.002. [35] 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. [36] Z. Jin and G. Yin, Numerical methods for optimal dividend payment and investment strategies of Markov-modulated jump diffusion models with regular and singular controls, Journal of Optimization Theory and Applications, 159 (2013), 246-271. doi: 10.1007/s10957-012-0263-7. [37] Z. Jin, H. Yang and G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica J. IFAC, 49 (2013), 2317-2329. doi: 10.1016/j.automatica.2013.04.043. [38] N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Springer, 1987. [39] H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, 2001. doi: 10.1007/978-1-4613-0007-6. [40] G. Leobacher, M. Szölgyenyi and S. Thonhauser, Bayesian dividend optimization and finite time ruin probabilities, Stochastic Models, 30 (2014), 216-249. doi: 10.1080/15326349.2014.900390. [41] R. C Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. [42] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer Science and Business Media, 2013. [43] J. Paulsen, Optimal dividend payouts for diffusions with solvency constraints, Finance and Stochastics, 7 (2003), 457-473. doi: 10.1007/s007800200098. [44] I. Pospelov and S. Radionov, Optimal dividend policy when cash surplus follows telegraph process, 2015. [45] P. E. Protter, Stochastic Integration and Differential Equations, Springer-Verlag, Berlin, 2005. doi: 10.1007/978-3-662-10061-5. [46] M. Reppen, J. Rochet and H. M. Soner. Optimal dividend policies with random profitability, arXiv preprint, arXiv: 1706.01813, 2017. [47] L. C. G. Rogers, Optimal Investment, Springer, 2013. doi: 10.1007/978-3-642-35202-7. [48] T. Rolski, H. Schmidli, V. Schmidt and J. Teugels, Stochastic Processes for Insurance and Finance, John Wiley & Sons, Ltd., Chichester, 1999. doi: 10.1002/9780470317044. [49] J. Sass and U. G. Haussmann, Optimizing the terminal wealth under partial information: The drift process as a continuous time markov chain, Finance and Stochastics, 8 (2004), 553-577. doi: 10.1007/s00780-004-0132-9. [50] Q. Song and C. Zhu, On singular control problems with state constraints and regime-switching: a viscosity solution approach, Automatica, 70 (2016), 66-73. doi: 10.1016/j.automatica.2016.03.017. [51] Q. Song, G. Yin and Z. Zhang, Numerical methods for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42 (2006), 1147-1157. doi: 10.1016/j.automatica.2006.03.016. [52] J. Stiglitz, Some aspects of the taxation of capital gains, Journal of Public Economics, 21 (1983), 257-294. doi: 10.3386/w1094. [53] M. Szölgyenyi, Dividend maximization in a hidden markov switching model, Statistics and Risk Modeling, 32 (2015), 143-158. doi: 10.1515/strm-2015-0019. [54] M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42. doi: 10.1007/s001860050001. [55] S. Tan, Z. Jin and G. Yin, Optimal dividend payment strategies with debt constraint in a hybrid regime-switching jump-diffusion model, Nonlinear Analysis: Hybrid Systems, 27 (2018), 141-156. doi: 10.1016/j.nahs.2017.08.007. [56] A. D. Wentzell, S. Chomet and K. L. Chung, A Course in the Theory of Stochastic Processes, McGraw-Hill International New York, 1981. [57] G. Xu and S. Shreve, A duality method for optimal consumption and investment under short-selling prohibition and general market coefficients, The Annals of Applied Probability, 2 (1992), 87-112. doi: 10.1214/aoap/1177005772. [58] D. Yao, H. Yang and R. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European Journal of Operational Research, 211 (2011), 568-576. doi: 10.1016/j.ejor.2011.01.015. [59] C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773. doi: 10.1016/j.insmatheco.2013.09.019. [60] K. C. Yiu, J. Liu, T. K. Siu and W. K. Ching, Optimal portfolios with regime switching and value-at-risk constraint, Automatica, 46 (2010), 979-989. doi: 10.1016/j.automatica.2010.02.027. [61] X. Zhang and T. K. Siu, On optimal proportional reinsurance and investment in a markovian regime-switching economy, Acta Mathematica Sinica, English Series, 28 (2012), 67-82. doi: 10.1007/s10114-012-9761-7. [62] Y. Zhao, R. Wang, D. Yao and P. Chen, Optimal dividends and capital injections in the dual model with a random time horizon, Journal of Optimization Theory and Applications, 167 (2015), 272-295. doi: 10.1007/s10957-014-0653-0. [63] L. Zheng, Optimal investment with high-watermark fee in a multi-dimensional jump diffusion model(doctoral dissertation), 2017. [64] J. Zhu, Singular optimal dividend control for the regime-switching Cramér-Lundberg model with credit and debit interest, Journal of Computational and Applied Mathematics, 257 (2014), 212-239. doi: 10.1016/j.cam.2013.08.033. [65] J. Zhu and F. Chen, Dividend optimization for regime-switching general diffusions, Insurance: Mathematics and Economics, 53 (2013), 439-456. doi: 10.1016/j.insmatheco.2013.07.006. [66] B. Zou and A. 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Comparison of optimal dividend amount
Comparison of optimal investment amount
Optimal dividend amount under bull and bear market
Optimal investment amount under bull and bear marke
Parameters setting in Example 1
 (Investor, Parameter) $\mu_1$ $\sigma_1$ $\mu_2$ $\sigma_2$ $p$ $\beta$ $\lambda$ $n$ Merton 0 0 0.06 0.3 0.33 0.15 0 - Financial Agent 0 0 0.06 0.3 0.33 0.15 0.2 10 Insurer 0.4 0.5 0.06 0.3 0.33 0.15 0.2 10
 (Investor, Parameter) $\mu_1$ $\sigma_1$ $\mu_2$ $\sigma_2$ $p$ $\beta$ $\lambda$ $n$ Merton 0 0 0.06 0.3 0.33 0.15 0 - Financial Agent 0 0 0.06 0.3 0.33 0.15 0.2 10 Insurer 0.4 0.5 0.06 0.3 0.33 0.15 0.2 10
Parameters setting in Example 2
 (State, Parameter) $\mu_1$ $\sigma_1$ $\mu_2$ $\sigma_2$ $p$ $\beta$ $\lambda$ $n$ Bull 0.4 0.5 0.06 0.3 0.3 0.05 0.2 10 Bear 0.3 0.5 0.03 0.3 0.3 0.05 0.2 10
 (State, Parameter) $\mu_1$ $\sigma_1$ $\mu_2$ $\sigma_2$ $p$ $\beta$ $\lambda$ $n$ Bull 0.4 0.5 0.06 0.3 0.3 0.05 0.2 10 Bear 0.3 0.5 0.03 0.3 0.3 0.05 0.2 10
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