### Optimal Reinsurance and Investment Strategies Under Mean-Variance Criteria: Partial and Full Information

ZHU Shihao, SHI Jingtao

1. School of Mathematics, Shandong University, Jinan 250100, China
• Received:2020-09-25 Revised:2021-07-26 Online:2022-08-25 Published:2022-08-02
• Supported by:
This paper was supported by National Key R&D Program of China under Grant No. 2018YFB1305400, the National Natural Science Foundations of China under Grant Nos. 11971266, 11831010, 11571205, and Shandong Provincial Natural Science Foundations under Grant Nos. ZR2020ZD24, ZR2019ZD42.

ZHU Shihao, SHI Jingtao. Optimal Reinsurance and Investment Strategies Under Mean-Variance Criteria: Partial and Full Information[J]. Journal of Systems Science and Complexity, 2022, 35(4): 1458-1479.

This paper is concerned with an optimal reinsurance and investment problem for an insurance firm under the criterion of mean-variance. The driving Brownian motion and the rate in return of the risky asset price dynamic equation cannot be directly observed. And the short-selling of stocks is prohibited. The problem is formulated as a stochastic linear-quadratic control problem where the control variables are constrained. Based on the separation principle and stochastic filtering theory, the partial information problem is solved. Efficient strategies and efficient frontier are presented in closed forms via solutions to two extended stochastic Riccati equations. As a comparison, the efficient strategies and efficient frontier are given by the viscosity solution to the HJB equation in the full information case. Some numerical illustrations are also provided.
 [1] Browne S, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 1995, 20(4): 937- 958.[2] Yang H L and Zhang L H, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 2005, 37(3): 615-634.[3] Promislow S D and Young V R, Minimizing the probability of ruin when claims follow Brownian motion with drift, North American Actuarial Journal, 2005, 9(3): 110-128.[4] Gu A and Li Z, Optimal reinsurance and investment strategies for insurers with regime-switching and state-dependent utility function, Journal of Systems Science and Complexity, 2016, 29(6): 1658-1682.[5] Li D, Rong X, and Zhao H, Optimal investment problem for an insurer and a reinsurer, Journal of Systems Science and Complexity, 2015, 28(6): 1326-1343.[6] Bai L H and Guo J Y, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 2008, 42(3): 968-975.[7] Liang Z B, Yuen K C, and Guo J Y, Optimal proportional reinsurance and investment in a stock market with Ornstein-Ohlenbeck process, Insurance: Mathematics and Economics, 2011, 49(2): 207-215.[8] Xu L, Zhang L M, and Yao D J, Optimal investment and reinsurance for an insurer under Markov-modulated financial market, Insurance: Mathematics and Economics, 2017, 74: 7-19.[9] Bielecki T R, Jin H Q, Pliska S R, et al., Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 2005, 15(2): 213-244.[10] Steinbach M C, Markowitz revisited: Mean-variance models in financial portfolio analysis, SIAM review, 2001, 43(1): 31-85.[11] MacLean L C, Zhao Y G, and Ziemba W T, Mean-variance versus expected utility in dynamic investment analysis, Computational Management Science, 2011, 8(1-2): 3-22.[12] Markowitz H, Portfolio selection, The Journal of Finance, 1952, 7(1): 77-91.[13] Li D and Ng W L, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 2000, 10(3): 387-406.[14] Zhou X Y and Li D, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 2000, 42(1): 19-33.[15] Li X, Zhou X Y, and Lim A E, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 2002, 40(5): 1540-1555.[16] Hu Y and Zhou X Y, Constrained stochastic LQ control with random coefficients, and application to portfolio selection, SIAM Journal on Control and Optimization, 2015, 44(2): 444-466.[17] Bai L H and Zhang H Y, Dynamic mean-variance problem with constrained risk control for the insurers, Mathematical Methods of Operations Research, 2008, 68(1): 181-205.[18] Bi J N, Meng Q B, and Zhang Y J, Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer, Annals of Operations Research, 2014, 212(1): 43-59.[19] Zhang N, Chen P, Jin Z, et al., Markowitz's mean-variance optimization with investment and constrained reinsurance, Journal of Industrial and Management Optimization, 2017, 13(1): 375- 397.[20] Di Nunno G and Ø ksendal B, Optimal portfolio, partial information and Malliavin calculus, Stochastics: An International Journal of Probability and Stochastics Processes, 2009, 81(3-4): 303-322.[21] Peng X C and Hu Y J, Optimal proportional reinsurance and investment under partial information, Insurance: Mathematics and Economics, 2013, 53(2): 416-428.[22] Wang G C and Wu Z, General maximum principles for partially observed risk-sensitive optimal control problems and applications to finance, Journal of Optimization Theory and Applications, 2009, 141(3): 677-700.[23] Huang J H, Wang G C, and Wu Z, Optimal premium policy of an insurance firm: Full and partial information, Insurance: Mathematics and Economics, 2010, 47(2): 208-215.[24] Al-Hussein A and Gherbal B, Necessary and sufficient optimality conditions for relaxed and strict control of forward-backward doubly SDEs with jumps under full and partial information, Journal of Systems Science and Complexity, 2020, 33(6): 1804-1846.[25] Pham H, Mean-variance hedging for partially observed drift processes, International Journal of Theoretical and Applied Finance, 2001, 4(2): 263-284.[26] Xiong J and Zhou X Y, Mean-variance portfolio selection under partial information, SIAM Journal on Control and Optimization, 2007, 46(1): 156-175.[27] Pang W K, Ni Y H, Li X, et al., Continuous-time mean-variance portfolio selection with partial information, Journal of Mathematical Finance, 2014, 4: 353-365.[28] Liang Z X and Song M, Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information, Insurance: Mathematics and Economics, 2015, 65: 66-76.[29] Cao J, Peng X C, and Hu Y J, Optimal time-consistent investment and reinsurance strategy for mean-variance insurers under the inside information, Acta Mathematicae Applicatae Sinica, English Series, 2016, 32(4): 1087-1100.[30] Li C L, Liu Z M, Wu J B, et al., The stochastic maximum principle for a jump-diffusion meanfield model involving impulse controls and applications in finance, Journal of Systems Science and Complexity, 2020, 33(1): 26-42.[31] Gennotte G, Optimal portfolio choice under incomplete information, The Journal of Finance, 1986, 41(3): 733-746.[32] Azcue P and Muler N, Optimal reinsurance and dividend distribution policies in the CramerLundberg model, Mathematical Finance, 2005, 15(2): 261-308.[33] Luo S, Wang M, and Zeng X, Optimal reinsurance: Minimize the expected time to reach a goal, Scandinavian Actuarial Journal, 2016, 8: 741-762.[34] Liptser R S and Shiryaev A N, Statistics of Random Processes: II. Applications, 2nd Edition, Springer-Verlag, New York, 1977.[35] Luenberger D, G, Optimization by Vector Space Methods, John Wiley and Sons, New York, 1969.[36] Yong J M and Zhou X Y, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.[37] Zhou X Y, Yong J M, and Li X J, Stochastic verification theorems within the framework of viscosity solutions, SIAM Journal on Control and Optimization, 1997, 35(1): 243-253.[38] Björk T, Khapko M, and Murgoci A, On time-inconsistent stochastic control in continuous time, Finance & Stochastics, 2017, 21: 331-360.[39] Björk T and Murgoci A, A general theory of Markovian time inconsistent stochastic control problems, Preprint, 2010, Electronic copy available at: http://ssrn.com/abstract=1694759.
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