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January  2022, 18(1): 75-93. doi: 10.3934/jimo.2020143

## Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition

 1 Sun Yat-sen Business School, Sun Yat-sen University, Guangzhou 510275, China 2 Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK 3 Business School, Jiangsu Normal University, Jiangsu 221116, China 4 Key Laboratory of Applied Statistics of MOE, School of Mathematics and Statistics, Northeast Normal University, Jilin 130024, China

* Corresponding author: Yin Li

Received  September 2018 Revised  February 2020 Published  January 2022 Early access  September 2020

In this study, under the criterion of maximizing the expected exponential utility of terminal wealth, the optimal proportional reinsurance and investment strategy for an insurer is examined with the compound Poisson claim process. To make the model more realistic, the price process of the risky asset is modelled by the Brownian motion risk model with dividends and transaction costs, where the instantaneous of investment return follows as a mean-reverting Ornstein-Uhlenbeck process. At the same time, the net profit condition and variance reinsurance premium principle are also considered. Using stochastic control theory, explicit expressions for the optimal policy and value function are derived, and various numerical examples are given to further demonstrate the effectiveness of the model.

Citation: Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial & Management Optimization, 2022, 18 (1) : 75-93. doi: 10.3934/jimo.2020143
##### References:
 [1] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar [2] L. Bai and J. Guo, Utility maximization with partial information: The HJB equation approach, Frontiers of Mathematics in China, 2 (2007), 527-537.  doi: 10.1007/s11464-007-0032-3.  Google Scholar [3] J. Bi and J. Cai, Optimal investment-reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets, Insurance: Mathematics and Economics, 85 (2019), 1-14.  doi: 10.1016/j.insmatheco.2018.11.007.  Google Scholar [4] S. Browne, Optimal investment policies for a firm with random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar [5] A. V. Baev and B. V. Bondarev, On the ruin probability of an insurance company dealing in a BS-market, Theory of Probability and Mathematical Statistics, 74 (2007), 11-23.  doi: 10.1090/S0094-9000-07-00693-X.  Google Scholar [6] W. Chen, D. Xiong and Z. Ye, Investment with sequence losses in an uncertain environment and mean-variance hedging, Stochastic Analysis and Applications, 25 (2007), 55-71.  doi: 10.1080/07362990601051872.  Google Scholar [7] A. Chen, T. Nguyen and M. Stadje, Optimal investment and dividend payment strategies with debt management and reinsurance, Insurance: Mathematics and Economics, 79 (2018), 194-209.  doi: 10.1016/j.insmatheco.2018.01.008.  Google Scholar [8] W. Fleming and H. Soner, Controlled Markov Process and Viscosity Solutions, Spring-Verlag, New York, 1993.  Google Scholar [9] H. Gerber, An Introduction to Mathematical Risk Theory, Heubner Foundation Monograph, 1979.  Google Scholar [10] J. Gaier, P. Grandits and W. Schachermeyer, Asymptotic ruin probabilities and optimal investment, The Annals of Applied Probability, 13 (2003), 1054-1076.  doi: 10.1214/aoap/1060202834.  Google Scholar [11] G. Guan, Z. Liang and J. Feng, Time-consistent proportional reinsurance and investment strategies under ambiguous environment, Insurance: Mathematics and Economics, 83 (2018), 122-133.  doi: 10.1016/j.insmatheco.2018.09.007.  Google Scholar [12] C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar [13] C. Hipp and M. Plum, Optimal investment for investors with state dependent income, and for insurers, Finance and Stochastics, 7 (2003), 299-321.  doi: 10.1007/s007800200095.  Google Scholar [14] C. Hipp and H. Schmidli, Asymptotics of ruin probabilities for controlled risk processes in the small claims case, Scandinavian Actuarial Journal, 2004 (2004), 321-335.  doi: 10.1080/03461230410000538.  Google Scholar [15] C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004.  Google Scholar [16] R. Kaas, M. Goovaerts, J. Dhaene and M. Denuit, Modern Actuarial Risk Theory Using R, Springer-Verlag Berlin Heidelberg, 2008. Google Scholar [17] I. Karatzas, Optimization problems in the theory of continuous trading, SIAM Journal on Control and Optimization, 27 (1989), 1221-1259.  doi: 10.1137/0327063.  Google Scholar [18] R. Kostadinova, Optimal investment for insurers when the stock price follows an exponential Lévy process, Insurance: Mathematics and Economics, 41 (2007), 250-263.  doi: 10.1016/j.insmatheco.2006.10.018.  Google Scholar [19] P. Lakner, Utility maximization with partial information, Stochastic Processes and their Applications, 56 (1995), 247-273.  doi: 10.1016/0304-4149(94)00073-3.  Google Scholar [20] P. Lakner, Optimal trading strategy for an investor: The case of partial information, Stochastic Processes and their Applications, 76 (1998), 77-97.  doi: 10.1016/S0304-4149(98)00032-5.  Google Scholar [21] D. Li, X. Rong and H. Zhao, Optimal investment problem with taxes, dividends and transaction costs under the constant elasticity of variance (CEV) model, Wseas Transactions on Mathematics, 12 (2013), 243-255.   Google Scholar [22] Z. Liang, Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion, Acta Mathematicae Applicatae Sinica (English Series), 23 (2007), 477-488.  doi: 10.1007/s10255-007-0387-y.  Google Scholar [23] Z. Liang, L. Bai and J. Guo, Optimal investment and proportional reinsurance with constrained control variables, Optimal Control Applications and Methods, 32 (2010), 587-608.  doi: 10.1002/oca.965.  Google Scholar [24] Z. Liang and J. Guo, Upper bound for ruin probabilities under optimal investment and proportional reinsurance, Applied Stochastic Models in Business and Industry, 24 (2008), 109-128.  doi: 10.1002/asmb.694.  Google Scholar [25] Z. Liang, K. Yuen and J. Guo, Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance: Mathematics and Economics, 49 (2011), 207-215.  doi: 10.1016/j.insmatheco.2011.04.005.  Google Scholar [26] K. Lindensj$\mathrm{\ddot{o}}$, Optimal investment and consumption under partial information, Mathematical Methods of Operations Research, 83 (2016), 87-107.  doi: 10.1007/s00186-015-0521-1.  Google Scholar [27] H. Loubergé and R. Watt, Insuring a risky investment project, Insurance: Mathematics and Economics, 42 (2008), 301-310.  doi: 10.1016/j.insmatheco.2007.03.003.  Google Scholar [28] S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444.  doi: 10.1016/j.insmatheco.2007.04.002.  Google Scholar [29] 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.  Google Scholar [30] D. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.  Google Scholar [31] R. Rishel, Optimal portfolio management with partial observation and power utility function, Stochastic Analysis, Control, Optimization and Applications, (1999), 605–619.  Google Scholar [32] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68.  doi: 10.1080/034612301750077338.  Google Scholar [33] W. Wang and X. Peng, Reinsurer's optimal reinsurance strategy with upper and lower premium constraints under distortion risk measures, Journal of Computational and Applied Mathematics, 315 (2017), 142-160.  doi: 10.1016/j.cam.2016.10.017.  Google Scholar [34] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar [35] Q. Zhao, J. Zhuo and J. Wei, Optimal investment and dividend payment strategies with debt management and reinsurance, Journal of Industrial and Management Optimization, 14 (2018), 1323-1348.  doi: 10.3934/jimo.2018009.  Google Scholar

show all references

##### References:
 [1] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002.  Google Scholar [2] L. Bai and J. Guo, Utility maximization with partial information: The HJB equation approach, Frontiers of Mathematics in China, 2 (2007), 527-537.  doi: 10.1007/s11464-007-0032-3.  Google Scholar [3] J. Bi and J. Cai, Optimal investment-reinsurance strategies with state dependent risk aversion and VaR constraints in correlated markets, Insurance: Mathematics and Economics, 85 (2019), 1-14.  doi: 10.1016/j.insmatheco.2018.11.007.  Google Scholar [4] S. Browne, Optimal investment policies for a firm with random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar [5] A. V. Baev and B. V. Bondarev, On the ruin probability of an insurance company dealing in a BS-market, Theory of Probability and Mathematical Statistics, 74 (2007), 11-23.  doi: 10.1090/S0094-9000-07-00693-X.  Google Scholar [6] W. Chen, D. Xiong and Z. Ye, Investment with sequence losses in an uncertain environment and mean-variance hedging, Stochastic Analysis and Applications, 25 (2007), 55-71.  doi: 10.1080/07362990601051872.  Google Scholar [7] A. Chen, T. Nguyen and M. Stadje, Optimal investment and dividend payment strategies with debt management and reinsurance, Insurance: Mathematics and Economics, 79 (2018), 194-209.  doi: 10.1016/j.insmatheco.2018.01.008.  Google Scholar [8] W. Fleming and H. Soner, Controlled Markov Process and Viscosity Solutions, Spring-Verlag, New York, 1993.  Google Scholar [9] H. Gerber, An Introduction to Mathematical Risk Theory, Heubner Foundation Monograph, 1979.  Google Scholar [10] J. Gaier, P. Grandits and W. Schachermeyer, Asymptotic ruin probabilities and optimal investment, The Annals of Applied Probability, 13 (2003), 1054-1076.  doi: 10.1214/aoap/1060202834.  Google Scholar [11] G. Guan, Z. Liang and J. Feng, Time-consistent proportional reinsurance and investment strategies under ambiguous environment, Insurance: Mathematics and Economics, 83 (2018), 122-133.  doi: 10.1016/j.insmatheco.2018.09.007.  Google Scholar [12] C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.  Google Scholar [13] C. Hipp and M. Plum, Optimal investment for investors with state dependent income, and for insurers, Finance and Stochastics, 7 (2003), 299-321.  doi: 10.1007/s007800200095.  Google Scholar [14] C. Hipp and H. Schmidli, Asymptotics of ruin probabilities for controlled risk processes in the small claims case, Scandinavian Actuarial Journal, 2004 (2004), 321-335.  doi: 10.1080/03461230410000538.  Google Scholar [15] C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004.  Google Scholar [16] R. Kaas, M. Goovaerts, J. Dhaene and M. Denuit, Modern Actuarial Risk Theory Using R, Springer-Verlag Berlin Heidelberg, 2008. Google Scholar [17] I. Karatzas, Optimization problems in the theory of continuous trading, SIAM Journal on Control and Optimization, 27 (1989), 1221-1259.  doi: 10.1137/0327063.  Google Scholar [18] R. Kostadinova, Optimal investment for insurers when the stock price follows an exponential Lévy process, Insurance: Mathematics and Economics, 41 (2007), 250-263.  doi: 10.1016/j.insmatheco.2006.10.018.  Google Scholar [19] P. Lakner, Utility maximization with partial information, Stochastic Processes and their Applications, 56 (1995), 247-273.  doi: 10.1016/0304-4149(94)00073-3.  Google Scholar [20] P. Lakner, Optimal trading strategy for an investor: The case of partial information, Stochastic Processes and their Applications, 76 (1998), 77-97.  doi: 10.1016/S0304-4149(98)00032-5.  Google Scholar [21] D. Li, X. Rong and H. Zhao, Optimal investment problem with taxes, dividends and transaction costs under the constant elasticity of variance (CEV) model, Wseas Transactions on Mathematics, 12 (2013), 243-255.   Google Scholar [22] Z. Liang, Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion, Acta Mathematicae Applicatae Sinica (English Series), 23 (2007), 477-488.  doi: 10.1007/s10255-007-0387-y.  Google Scholar [23] Z. Liang, L. Bai and J. Guo, Optimal investment and proportional reinsurance with constrained control variables, Optimal Control Applications and Methods, 32 (2010), 587-608.  doi: 10.1002/oca.965.  Google Scholar [24] Z. Liang and J. Guo, Upper bound for ruin probabilities under optimal investment and proportional reinsurance, Applied Stochastic Models in Business and Industry, 24 (2008), 109-128.  doi: 10.1002/asmb.694.  Google Scholar [25] Z. Liang, K. Yuen and J. Guo, Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance: Mathematics and Economics, 49 (2011), 207-215.  doi: 10.1016/j.insmatheco.2011.04.005.  Google Scholar [26] K. Lindensj$\mathrm{\ddot{o}}$, Optimal investment and consumption under partial information, Mathematical Methods of Operations Research, 83 (2016), 87-107.  doi: 10.1007/s00186-015-0521-1.  Google Scholar [27] H. Loubergé and R. Watt, Insuring a risky investment project, Insurance: Mathematics and Economics, 42 (2008), 301-310.  doi: 10.1016/j.insmatheco.2007.03.003.  Google Scholar [28] S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444.  doi: 10.1016/j.insmatheco.2007.04.002.  Google Scholar [29] 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.  Google Scholar [30] D. Promislow and V. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 9 (2005), 109-128.  doi: 10.1080/10920277.2005.10596214.  Google Scholar [31] R. Rishel, Optimal portfolio management with partial observation and power utility function, Stochastic Analysis, Control, Optimization and Applications, (1999), 605–619.  Google Scholar [32] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1 (2001), 55-68.  doi: 10.1080/034612301750077338.  Google Scholar [33] W. Wang and X. Peng, Reinsurer's optimal reinsurance strategy with upper and lower premium constraints under distortion risk measures, Journal of Computational and Applied Mathematics, 315 (2017), 142-160.  doi: 10.1016/j.cam.2016.10.017.  Google Scholar [34] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009.  Google Scholar [35] Q. Zhao, J. Zhuo and J. Wei, Optimal investment and dividend payment strategies with debt management and reinsurance, Journal of Industrial and Management Optimization, 14 (2018), 1323-1348.  doi: 10.3934/jimo.2018009.  Google Scholar
The optimal proportional level ${q^*}\left( t \right)$ with $n = 0.2, n = 0.5, n = 1$, respectively
The optimal proportional level ${q^*}\left( t \right)$ with $\Lambda = 1, \Lambda = 1.2, \Lambda = 1.5$, respectively
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