# American Institute of Mathematical Sciences

April  2018, 14(2): 653-671. doi: 10.3934/jimo.2017067

## Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer

 1 School of Science, Dalian Jiaotong University, Dalian 116028, China 2 Department of Mathematics, The George Washington University, Washington DC 20052, USA 3 School of Mathematical Sciences, Dalian University of Technology, Dalian 116023, China 4 Department of Mathematics, Loyola Marymount University, Los Angeles CA 90045, USA

* Corresponding author. E-mail address: wanglei@dlut.edu.cn

The reviewing process was handled by Changjun Yu

Received  April 2016 Revised  December 2016 Published  June 2017

Fund Project: This work was supported by the National Natural Science Foundation for the Youth of China (Grants 11301081, 11401073), the Science Research Project of Educational Department of Liaoning Province of China (Grants. L2014188, L2015097 and L2014186), the Research Funding for Doctor Start-Up Program of Liaoning Province (Grant 201601245), the Fundamental Research Funds for Central Universities in China (Grant DUT15LK25), the Simons Foundation through Grant No. 357963 (Y.Z.), a start-up grant from the George Washington University (Y.Z.), Loyola Marymount University CSE continuing Faculty Research grant (Y.M.), and a start-up grant from Loyola Marymount University (Y.M.).

We study an optimal investment and dividend problem of an insurer, where the aggregate insurance claims process is modeled by a pure jump Lévy process. We allow the management of the dividend payment policy and the investment of surplus in a continuous-time financial market, which is composed of a risk free asset and a risky asset. The information available to the insurer is partial information. We generalize this problem as a partial information regular-singular stochastic control problem, where the control variable consists of regular control and singular control. Then maximum principles are established to give sufficient and necessary optimality conditions for the solutions of the regular-singular control problem. Finally we apply the maximum principles to solve the investment and dividend problem of an insurer.

Citation: Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067
##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge university press, New York, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar [2] F. Avram, Z. Palmowski and M. R. Pistorius, On Gerber-Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function, Ann. Appl. Probab., 25 (2015), 1868-1935.  doi: 10.1214/14-AAP1038.  Google Scholar [3] P. Azcue and N. Muler, Optimal investment policy and dividend payment strategy in an insurance company, Ann. Appl. Probab., 20 (2010), 1253-1302.  doi: 10.1214/09-AAP643.  Google Scholar [4] F. Baghery and B. Oksendal, A maximum principle for stochastic control with partial information, Stoch. Anal. Appl., 25 (2007), 705-717.  doi: 10.1080/07362990701283128.  Google Scholar [5] M. Belhaj, Optimal dividend payments when cash reserves follow a jump-diffusion process, Math. Finance, 20 (2010), 313-325.  doi: 10.1111/j.1467-9965.2010.00399.x.  Google Scholar [6] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar [7] G. Cheng, R. Wang and K. Fan, Optimal risk and dividend control of an insurance company with exponential premium principle and liquidation value, Stochastics, 88 (2016), 904-926.  doi: 10.1080/17442508.2016.1163362.  Google Scholar [8] B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443.   Google Scholar [9] R. J. Elliott and T. K. Siu, A stochastic differential game for optimal investment of an insurer with regime switching, Quant. Finance, 11 (2011), 365-380.  doi: 10.1080/14697681003591704.  Google Scholar [10] W. Guo, Optimal portfolio choice for an insurer with loss aversion, Insurance Math. Econom., 58 (2014), 217-222.  doi: 10.1016/j.insmatheco.2014.07.004.  Google Scholar [11] M. Hafayed, M. Ghebouli, S. Boukaf and Y. Shi, Partial information optimal control of mean-field forward-backward stochastic system driven by Teugels martingales with applications, Neurocomputing, 200 (2016), 11-21.  doi: 10.1016/j.neucom.2016.03.002.  Google Scholar [12] B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quant. Finance, 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007.  Google Scholar [13] Z. Jin, H. Yang and G. G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica, 49 (2013), 2317-2329.  doi: 10.1016/j.automatica.2013.04.043.  Google Scholar [14] 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, J. Optim. Theory Appl., 159 (2013), 246-271.  doi: 10.1007/s10957-012-0263-7.  Google Scholar [15] X. Lin, C. Zhang and T. K. Siu, Stochastic differential portfolio games for an insurer in a jump-diffusion risk process, Math. Methods Oper. Res., 75 (2012), 83-100.  doi: 10.1007/s00186-011-0376-z.  Google Scholar [16] C. S. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, N. Am. Actuar. J., 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.  Google Scholar [17] J. Liu, K. F. C. Yiu and T. K. Siu, Optimal investment of an insurer with regime-switching and risk constraint, Scand. Actuar. J., 7 (2014), 583-601.  doi: 10.1080/03461238.2012.750621.  Google Scholar [18] E. Marciniak and Z. Palmowski, On the optimal dividend problem for insurance risk models with surplus-dependent premiums, J. Optim. Theory Appl., 168 (2016), 723-742.  doi: 10.1007/s10957-015-0755-3.  Google Scholar [19] B. Oksendal and A. Sulém, Singular stochastic control and optimal stopping with partial information of Itô-Lévy processes, SIAM J. Control Optim., 50 (2012), 2254-2287.  doi: 10.1137/100793931.  Google Scholar [20] H. Markovitz, Portfolio selection*, J. Finance, 7 (1952), 77-91.   Google Scholar [21] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar [22] M. I. Taksar, Optimal risk and dividend distribution control models for an insurance company, Math. Methods Oper. Res., 51 (2000), 1-42.  doi: 10.1007/s001860050001.  Google Scholar [23] Y. Wang, A. Song and E. Feng, A maximum principle via Malliavin calculus for combined stochastic control and impulse control of forward-backward systems, Asian J. Control, 17 (2015), 1798-1809.  doi: 10.1002/asjc.1097.  Google Scholar [24] F. Zhang, Stochastic maximum principle for mixed regular-singular control problems of forward-backward systems, J. Syst. Sci. Complex., 26 (2013), 886-901.  doi: 10.1007/s11424-013-0287-6.  Google Scholar

show all references

##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge university press, New York, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar [2] F. Avram, Z. Palmowski and M. R. Pistorius, On Gerber-Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function, Ann. Appl. Probab., 25 (2015), 1868-1935.  doi: 10.1214/14-AAP1038.  Google Scholar [3] P. Azcue and N. Muler, Optimal investment policy and dividend payment strategy in an insurance company, Ann. Appl. Probab., 20 (2010), 1253-1302.  doi: 10.1214/09-AAP643.  Google Scholar [4] F. Baghery and B. Oksendal, A maximum principle for stochastic control with partial information, Stoch. Anal. Appl., 25 (2007), 705-717.  doi: 10.1080/07362990701283128.  Google Scholar [5] M. Belhaj, Optimal dividend payments when cash reserves follow a jump-diffusion process, Math. Finance, 20 (2010), 313-325.  doi: 10.1111/j.1467-9965.2010.00399.x.  Google Scholar [6] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937.  Google Scholar [7] G. Cheng, R. Wang and K. Fan, Optimal risk and dividend control of an insurance company with exponential premium principle and liquidation value, Stochastics, 88 (2016), 904-926.  doi: 10.1080/17442508.2016.1163362.  Google Scholar [8] B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443.   Google Scholar [9] R. J. Elliott and T. K. Siu, A stochastic differential game for optimal investment of an insurer with regime switching, Quant. Finance, 11 (2011), 365-380.  doi: 10.1080/14697681003591704.  Google Scholar [10] W. Guo, Optimal portfolio choice for an insurer with loss aversion, Insurance Math. Econom., 58 (2014), 217-222.  doi: 10.1016/j.insmatheco.2014.07.004.  Google Scholar [11] M. Hafayed, M. Ghebouli, S. Boukaf and Y. Shi, Partial information optimal control of mean-field forward-backward stochastic system driven by Teugels martingales with applications, Neurocomputing, 200 (2016), 11-21.  doi: 10.1016/j.neucom.2016.03.002.  Google Scholar [12] B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quant. Finance, 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007.  Google Scholar [13] Z. Jin, H. Yang and G. G. Yin, Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections, Automatica, 49 (2013), 2317-2329.  doi: 10.1016/j.automatica.2013.04.043.  Google Scholar [14] 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, J. Optim. Theory Appl., 159 (2013), 246-271.  doi: 10.1007/s10957-012-0263-7.  Google Scholar [15] X. Lin, C. Zhang and T. K. Siu, Stochastic differential portfolio games for an insurer in a jump-diffusion risk process, Math. Methods Oper. Res., 75 (2012), 83-100.  doi: 10.1007/s00186-011-0376-z.  Google Scholar [16] C. S. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, N. Am. Actuar. J., 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.  Google Scholar [17] J. Liu, K. F. C. Yiu and T. K. Siu, Optimal investment of an insurer with regime-switching and risk constraint, Scand. Actuar. J., 7 (2014), 583-601.  doi: 10.1080/03461238.2012.750621.  Google Scholar [18] E. Marciniak and Z. Palmowski, On the optimal dividend problem for insurance risk models with surplus-dependent premiums, J. Optim. Theory Appl., 168 (2016), 723-742.  doi: 10.1007/s10957-015-0755-3.  Google Scholar [19] B. Oksendal and A. Sulém, Singular stochastic control and optimal stopping with partial information of Itô-Lévy processes, SIAM J. Control Optim., 50 (2012), 2254-2287.  doi: 10.1137/100793931.  Google Scholar [20] H. Markovitz, Portfolio selection*, J. Finance, 7 (1952), 77-91.   Google Scholar [21] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3 (1971), 373-413.  doi: 10.1016/0022-0531(71)90038-X.  Google Scholar [22] M. I. Taksar, Optimal risk and dividend distribution control models for an insurance company, Math. Methods Oper. Res., 51 (2000), 1-42.  doi: 10.1007/s001860050001.  Google Scholar [23] Y. Wang, A. Song and E. Feng, A maximum principle via Malliavin calculus for combined stochastic control and impulse control of forward-backward systems, Asian J. Control, 17 (2015), 1798-1809.  doi: 10.1002/asjc.1097.  Google Scholar [24] F. Zhang, Stochastic maximum principle for mixed regular-singular control problems of forward-backward systems, J. Syst. Sci. Complex., 26 (2013), 886-901.  doi: 10.1007/s11424-013-0287-6.  Google Scholar
 [1] Yiling Chen, Baojun Bian. optimal investment and dividend policy in an insurance company: A varied bound for dividend rates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5083-5105. doi: 10.3934/dcdsb.2019044 [2] Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581 [3] Shuaiqi Zhang, Jie Xiong, Xin Zhang. Optimal investment problem with delay under partial information. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020001 [4] Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018 [5] Linlin Tian, Xiaoyi Zhang, Yizhou Bai. Optimal dividend of compound poisson process under a stochastic interest rate. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019047 [6] Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022 [7] Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195 [8] Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499 [9] Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100 [10] Dingjun Yao, Kun Fan. Optimal risk control and dividend strategies in the presence of two reinsurers: Variance premium principle. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1055-1083. doi: 10.3934/jimo.2017090 [11] Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298 [12] Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174 [13] Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161 [14] Qian Zhao, Zhuo Jin, Jiaqin Wei. Optimal investment and dividend payment strategies with debt management and reinsurance. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1323-1348. doi: 10.3934/jimo.2018009 [15] Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415 [16] Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019080 [17] Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial & Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585 [18] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77 [19] Jingzhen Liu, Ka-Fai Cedric Yiu, Tak Kuen Siu, Wai-Ki Ching. Optimal insurance in a changing economy. Mathematical Control & Related Fields, 2014, 4 (2) : 187-202. doi: 10.3934/mcrf.2014.4.187 [20] 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, 2020, 16 (1) : 325-356. doi: 10.3934/jimo.2018154

2018 Impact Factor: 1.025

Article outline

[Back to Top]