# 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [20] H. Markovitz, Portfolio selection*, J. Finance, 7 (1952), 77-91. [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. [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. [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. [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.

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##### References:
 [1] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge university press, New York, 2009. doi: 10.1017/CBO9780511809781. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [20] H. Markovitz, Portfolio selection*, J. Finance, 7 (1952), 77-91. [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. [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. [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. [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.
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