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Maximizing expected terminal utility of an insurer with high gain tax by investment and reinsurance. (English) Zbl 1448.91269
Summary: This paper investigates optimal investment and proportional reinsurance policies for an insurer who subjects to pay high gain tax. The surplus process of the insurer and the return process of the financial market are both modulated by the external macroeconomic environment. The dynamic of the external macroeconomic environment is specified by a Markov chain with finite states. Once the insurer’s accumulated profits attain a new maximum, they have to pay high gain tax. The objective of the insurer is to maximize the expected terminal utility by investment and reinsurance. The controlled wealth process of the insurer turned out to be a controlled jump diffusion process with reflections and Markov regime switching. By the weak dynamic programming principle (WDPP), we prove that the value function is the unique viscosity solution to the coupled Hamilton-Jacob-Bellman (HJB) equations with first derivative boundary constraints. By the Markov chain approximating method for the HJB equations, we construct a numerical scheme for approximating the viscosity solution to the coupled HJB equations. Two numerical examples are presented to illustrate the impact of both high gain tax and regime switching on the optimal policies.

##### MSC:
 91G05 Actuarial mathematics 91B64 Macroeconomic theory (monetary models, models of taxation) 60J28 Applications of continuous-time Markov processes on discrete state spaces 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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##### References:
 [1] Schmidli, H., On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 890-907 (2002) · Zbl 1021.60061 [2] Taksar, M. I.; Markussen, C., Optimal dynamic reinsurance policies for large insurance portfolios, Finance Stoch., 7, 1, 97-121 (2003) · Zbl 1066.91052 [3] Browne, S., Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20, 4, 937-958 (1995) · Zbl 0846.90012 [4] Hipp, C.; Plum, M., Optimal investment for investors with state dependent income, and for insurers, Finance Stoch., 7, 3, 299-321 (2003) · Zbl 1069.91051 [5] Gaier, J.; Grandits, P.; Schachermayer, W., Asymptotic ruin probabilities and optimal investment, Ann. Appl. Probab., 1054-1076 (2003) · Zbl 1046.62113 [6] Yang, H.; Zhang, L., Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37, 3, 615-634 (2005) · Zbl 1129.91020 [7] Luo, S.; Taksar, M.; Tsoi, A., On reinsurance and investment for large insurance portfolios, Insurance Math. Econom., 42, 1, 434-444 (2008) · Zbl 1141.91532 [8] Azcue, P.; Muler, N., Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints, Insurance Math. Econom., 44, 1, 26-34 (2009) · Zbl 1156.91391 [9] Zhang, X.; Siu, T. K., Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Math. Econom., 45, 1, 81-88 (2009) · Zbl 1231.91257 [10] Yin, C.; Wang, C., The perturbed compound Poisson risk process with investment and debit interest, Methodol. Comput. Appl. Probab., 12, 3, 391-413 (2010) · Zbl 1231.91255 [11] Zeng, Y.; Li, Z., Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49, 1, 145-154 (2011) · Zbl 1218.91167 [12] He, L.; Liang, Z., Optimal dynamic asset allocation strategy for ELA scheme of DC pension plan during the distribution phase, Insurance Math. Econom., 52, 2, 404-410 (2013) · Zbl 1284.91521 [13] Zhao, H.; Rong, X.; Zhao, Y., Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the heston model, Insurance Math. Econom., 53, 3, 504-514 (2013) · Zbl 1290.91106 [14] Yin, C.; Wen, Y., An extension of paulsen-gjessings risk model with stochastic return on investments, Insurance Math. Econom., 52, 3, 469-476 (2013) · Zbl 1284.91281 [15] Zhao, Y.; Wang, R.; Yao, D.; Chen, P., Optimal dividends and capital injections in the dual model with a random time horizon, J. Optim. Theory Appl., 167, 1, 272-295 (2015) · Zbl 1341.49021 [16] Peng, X.; Chen, F.; Hu, Y., Optimal investment, consumption and proportional reinsurance under model uncertainty, Insurance Math. Econom., 59, 222-234 (2014) · Zbl 1306.91131 [17] Shen, Y.; Zeng, Y., Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process, Insurance Math. Econom., 62, 118-137 (2015) · Zbl 1318.91123 [18] Li, D.; Rong, X.; Zhao, H., The optimal investment problem for an insurer and a reinsurer under the constant elasticity of variance model, IMA J. Manag. Math., 27, 2, 255-280 (2016) · Zbl 1433.91137 [19] Zhou, M.; Yuen, K. C.; Yin, C.-c., Optimal investment and premium control in a nonlinear diffusion model, Acta Math. Appl. Sin. Engl. Ser., 33, 4, 945-958 (2017) · Zbl 1402.91220 [20] Bai, L.; Guo, J., Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42, 3, 968-975 (2008) · Zbl 1147.93046 [21] Yiu, K.-F. C.; Liu, J.; Siu, T. K.; Ching, W.-K., Optimal portfolios with regime switching and value-at-risk constraint, Automatica, 46, 6, 979-989 (2010) · Zbl 1189.91199 [22] Huang, Y.; Yang, X.; Zhou, J., Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, J. Comput. Appl. Math., 296, 443-461 (2016) · Zbl 1331.91097 [23] Zhang, X.; Meng, H.; Zeng, Y., Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling, Insurance Math. Econom., 67, 125-132 (2016) · Zbl 1348.91193 [24] Dammon, R. M.; Spatt, C. S.; Zhang, H. H., Optimal consumption and investment with capital gains taxes, Rev. Financ. Stud., 14, 3, 583-616 (2001) [25] Janecek, K.; Sîrbu, M., Optimal investment with high-watermark performance fee, SIAM J. Control Optim., 50, 2, 790-819 (2012) · Zbl 1248.91092 [26] Albrecher, H.; Hipp, C., Lundbergs risk process with tax, Bl. DGVFM, 28, 1, 13-28 (2007) · Zbl 1119.62103 [27] Wei, J.; Yang, H.; Wang, R., On the Markov-modulated insurance risk model with tax, Bl. DGVFM, 31, 1, 65-78 (2010) · Zbl 1195.91071 [28] Zhang, Z.; Cheung, E. C.; Yang, H., Lévy insurance risk process with Poissonian taxation, Scand. Actuar. J., 2017, 1, 51-87 (2017) · Zbl 1401.91216 [29] Bäuerle, N.; Rieder, U., Portfolio optimization with Markov-modulated stock prices and interest rates, IEEE Trans. Automat. Control, 49, 3, 442-447 (2004) · Zbl 1366.91135 [30] Sass, J.; Haussmann, U. G., Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain, Finance Stoch., 8, 4, 553-577 (2004) · Zbl 1063.91040 [31] Ng, A. C.; Yang, H., On the joint distribution of surplus before and after ruin under a Markovian regime switching model, Stochastic Process. Appl., 116, 2, 244-266 (2006) · Zbl 1093.60051 [32] Zhu, J., Optimal dividend control for a generalized risk model with investment incomes and debit interest, Scand. Actuar. J., 2013, 2, 140-162 (2013) [33] Fu, J.; Wei, J.; Yang, H., Portfolio optimization in a regime-switching market with derivatives, European J. Oper. Res., 233, 1, 184-192 (2014) · Zbl 1339.91108 [34] Zou, B., Stochastic Control in Optimal Insurance and Investment with Regime Switching (2014), University of Alberta, (Ph.D. thesis) [35] Hindy, A.; Huang, C.-f.; Zhu, S. H., Numerical analysis of a free-boundary singular control problem in financial economics, J. Econom. Dynam. Control, 21, 2, 297-327 (1997) · Zbl 0879.90021 [36] Song, Q., Convergence of Markov chain approximation on generalized HJB equation and its applications, Automatica, 44, 3, 761-766 (2008) · Zbl 1283.93319 [37] Xu, L.; Yao, D.; Cheng, G., Optimal investment and dividend for an insurer under a Markov regime switching market with high gain tax, J. Ind. Manag. Optim., 217-251 (2018) [38] Elliott, R. J.; Aggoun, L.; Moore, J. B., Hidden Markov Models (1994), Springer [39] Grandell, J., Aspects of risk theory (1991), Springer · Zbl 0717.62100 [40] Fleming, W. H.; Soner, H. M.; Soner, H. M.; Soner, H. M., Controlled Markov processes and viscosity solutions, vol. 25 (2006), Springer · Zbl 1105.60005 [41] Protter, P. E., Stochastic Integration and Differential Equations: Version 2.1, vol. 21 (2004), Springer [42] Bouchard, B.; Touzi, N., Weak dynamic programming principle for viscosity solutions, SIAM J. Control Optim., 49, 3, 948-962 (2011) · Zbl 1228.49028 [43] Benth, F. E.; Karlsen, K. H.; Reikvam, K., Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs, Stochastics, 74, 3-4, 517-569 (2002) · Zbl 1035.91027 [44] Crandall, M. G.; Ishii, H.; Lions, P.-L., Users guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27, 1, 1-67 (1992) · Zbl 0755.35015 [45] Bayraktar, E.; Dolinsky, Y.; Guo, J., Continuity of utility maximization under weak convergence (2018), Available at SSRN 3278294 [46] Kushner, H. J.; Dupuis, P., Numerical methods for stochastic control problems in continuous time, vol. 24 (2001), Springer [47] Song, Q.; Yin, G.; Zhang, Z., Numerical methods for controlled regime-switching diffusions and regime-switching jump diffusions, Automatica, 42, 7, 1147-1157 (2006) · Zbl 1117.93370 [48] Jin, Z.; Yin, G.; Wu, F., Optimal reinsurance strategies in regime-switching jump diffusion models: Stochastic differential game formulation and numerical methods, Insurance Math. Econom., 53, 3, 733-746 (2013) · Zbl 1290.91090 [49] Barles, G.; Souganidis, P. E., Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4, 3, 271-283 (1991) · Zbl 0729.65077
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