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## Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market

 1 School of Statistics, Chengdu University of Information Technology, Chengdu, 610103, China 2 Sichuan Administration Institute, Chengdu, 610072, China 3 School of hospitality Management, Shanghai Business School, Shanghai, 200235, China

*Corresponding author: Peimin Chen

Received  February 2021 Revised  February 2022 Early access April 2022

Fund Project: The Sheng Li is supported by the project of the Chengdu University of Information Technology Introducing Talents to Launch Scientific Research (KYTZ202193)

In this paper, we consider an optimal mean-variance investment and reinsurance problem with delay and Common Shock Dependence. An insurer can control the claim risk by purchasing proportional reinsurance. He/she invests his/her wealth on a risk-free asset and a risky asset, which follows the jump-diffusion process. By introducing a capital flow related to the historical performance of the insurer, the wealth process described by a stochastic differential equation with delay is obtained. By stochastic linear-quadratic control theory and stochastic control theory with delay, we achieve the explicit expression of the optimal strategy and value function in the framework of the viscosity solution. Furthermore, an efficient strategy and its efficient frontier are derived by Lagrange dual method. Finally, we analyze the influence of the parameters of our model on the efficient frontier by a numerical example.

Citation: Sheng Li, Wei Yuan, Peimin Chen. Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022068
##### References:
 [1] C. A, Y. Lai and S. Yi, Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the CEV model, J. Comput. Appl. Math, 342 (2018), 317-336.  doi: 10.1016/j.cam.2018.03.035. [2] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005. [3] L. Bai, J. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance Math. Econom., 53 (2013), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008. [4] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [5] L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4. [6] N. Bäuerle, Benchmark and mean-variance problems for insurers, Math. Methods Oper. Res., 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1. [7] J. Bi, J. Cai and Y. Zeng, Equilibrium reinsurance-investment strategies with partial information and common shock dependence, Ann. Oper. Res., 307 (2021), 1-24.  doi: 10.1007/s10479-021-04317-4. [8] J. Bi and K. Chen, Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles, RAIRO Oper. Res., 53 (2019), 179-206.  doi: 10.1051/ro/2019010. [9] J. Bi and J. Guo, Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, J. Optim. Theory Appl., 157 (2013), 252-275.  doi: 10.1007/s10957-012-0138-y. [10] J. Bi, Z. Liang and F. Xu, Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence, Insurance Math. Econom., 70 (2016), 245-258.  doi: 10.1016/j.insmatheco.2016.06.012. [11] J. Bi, Q. Meng and Y. Zhang, Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer, Ann. Oper. Res., 212 (2014), 43-59.  doi: 10.1007/s10479-013-1338-z. [12] N. Branger and L. S. Larsen, Robust portfolio choice with uncertainty about jump and diffusion risk, J. Banking and Finance, 37 (2013), 5036-5047. [13] 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. [14] Y. Cao and N. Wan, Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation, Insurance Math. Econom., 45 (2009), 157-162.  doi: 10.1016/j.insmatheco.2009.05.006. [15] M. H. Chang, T. Pang and Y. Yang, A stochastic portfolio optimization model with bounded memory, Math. Oper. Res., 36 (2011), 604-619.  doi: 10.1287/moor.1110.0508. [16] I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics Stochastics Rep., 71 (2000), 69-89.  doi: 10.1080/17442500008834259. [17] S. Federico, A stochastic control problem with delay arising in a pension fund model, Finance Stoch., 15 (2011), 421-459.  doi: 10.1007/s00780-010-0146-4. [18] C. Hipp and M. Plum, Optimal investment for insurers, Insurance Math. Econom., 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4. [19] B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models with transaction costs, Insurance Math. Econom., 22 (1998), 41-51.  doi: 10.1016/S0167-6687(98)00007-9. [20] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Math. Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [21] S. Li, Optimal time-consistent investment-reinsurance strategy for state-dependent risk aversion with delay and common shocks, Communications in Statistics-Theory and Methods, (2021), 1-38. [22] S. Li and Z. Qiu, Equilibrium investment-reinsurance strategy with delay and common shock dependence under heston's sv model, Optimization, (2021), 1-32. [23] Z. Liang and K. C. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scand. Actuar. J., 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899. [24] J. Liu, F. A. Longstaff and J. Pan, Dynamic asset allocation with event risk, J. Finance, 58 (2003), 231-259. [25] H. M. Markowitz, Portfolio Selection, Cowles Foundation for Research in Economics at Yale University, Monograph 16 John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London 1959. [26] Z. Ming, Z. Liang and C. Zhang, Optimal mean-variance reinsurance with common shock dependence, ANZIAM J., 58 (2016), 162-181.  doi: 10.1017/S144618111600016X. [27] S. E. A. Mohammed, Stochastic functional differential equations, Research Notes in Mathematics, 99. Pitman (Advanced Publishing Program), Boston, MA, 1984. [28] B. Øksendal and S. Agnès, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, (2001), 64–79. [29] Y. Shen and Y. Zeng, Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach, Insurance Math. Econom., 57 (2014), 1-12.  doi: 10.1016/j.insmatheco.2014.04.004. [30] H. M. Soner and W. H. Fleming, Controlled Markov Processes and Viscosity Solutions, 2$^nd$ edition, Springer-Verlag, New York, 2006. [31] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [32] K. C. Yuen, Z. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance Math. Econom., 64 (2015), 1-13.  doi: 10.1016/j.insmatheco.2015.04.009. [33] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33.  doi: 10.1007/s002450010003.

show all references

##### References:
 [1] C. A, Y. Lai and S. Yi, Optimal excess-of-loss reinsurance and investment problem with delay and jump-diffusion risk process under the CEV model, J. Comput. Appl. Math, 342 (2018), 317-336.  doi: 10.1016/j.cam.2018.03.035. [2] C. A and Z. Li, Optimal investment and excess-of-loss reinsurance problem with delay for an insurer under Heston's SV model, Insurance Math. Econom., 61 (2015), 181-196.  doi: 10.1016/j.insmatheco.2015.01.005. [3] L. Bai, J. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance Math. Econom., 53 (2013), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008. [4] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [5] L. Bai and H. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Math. Methods Oper. Res., 68 (2008), 181-205.  doi: 10.1007/s00186-007-0195-4. [6] N. Bäuerle, Benchmark and mean-variance problems for insurers, Math. Methods Oper. Res., 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1. [7] J. Bi, J. Cai and Y. Zeng, Equilibrium reinsurance-investment strategies with partial information and common shock dependence, Ann. Oper. Res., 307 (2021), 1-24.  doi: 10.1007/s10479-021-04317-4. [8] J. Bi and K. Chen, Optimal investment-reinsurance problems with common shock dependent risks under two kinds of premium principles, RAIRO Oper. Res., 53 (2019), 179-206.  doi: 10.1051/ro/2019010. [9] J. Bi and J. Guo, Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, J. Optim. Theory Appl., 157 (2013), 252-275.  doi: 10.1007/s10957-012-0138-y. [10] J. Bi, Z. Liang and F. Xu, Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence, Insurance Math. Econom., 70 (2016), 245-258.  doi: 10.1016/j.insmatheco.2016.06.012. [11] J. Bi, Q. Meng and Y. Zhang, Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer, Ann. Oper. Res., 212 (2014), 43-59.  doi: 10.1007/s10479-013-1338-z. [12] N. Branger and L. S. Larsen, Robust portfolio choice with uncertainty about jump and diffusion risk, J. Banking and Finance, 37 (2013), 5036-5047. [13] 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. [14] Y. Cao and N. Wan, Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation, Insurance Math. Econom., 45 (2009), 157-162.  doi: 10.1016/j.insmatheco.2009.05.006. [15] M. H. Chang, T. Pang and Y. Yang, A stochastic portfolio optimization model with bounded memory, Math. Oper. Res., 36 (2011), 604-619.  doi: 10.1287/moor.1110.0508. [16] I. Elsanosi, B. Øksendal and A. Sulem, Some solvable stochastic control problems with delay, Stochastics Stochastics Rep., 71 (2000), 69-89.  doi: 10.1080/17442500008834259. [17] S. Federico, A stochastic control problem with delay arising in a pension fund model, Finance Stoch., 15 (2011), 421-459.  doi: 10.1007/s00780-010-0146-4. [18] C. Hipp and M. Plum, Optimal investment for insurers, Insurance Math. Econom., 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4. [19] B. Højgaard and M. Taksar, Optimal proportional reinsurance policies for diffusion models with transaction costs, Insurance Math. Econom., 22 (1998), 41-51.  doi: 10.1016/S0167-6687(98)00007-9. [20] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Math. Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [21] S. Li, Optimal time-consistent investment-reinsurance strategy for state-dependent risk aversion with delay and common shocks, Communications in Statistics-Theory and Methods, (2021), 1-38. [22] S. Li and Z. Qiu, Equilibrium investment-reinsurance strategy with delay and common shock dependence under heston's sv model, Optimization, (2021), 1-32. [23] Z. Liang and K. C. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scand. Actuar. J., 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899. [24] J. Liu, F. A. Longstaff and J. Pan, Dynamic asset allocation with event risk, J. Finance, 58 (2003), 231-259. [25] H. M. Markowitz, Portfolio Selection, Cowles Foundation for Research in Economics at Yale University, Monograph 16 John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London 1959. [26] Z. Ming, Z. Liang and C. Zhang, Optimal mean-variance reinsurance with common shock dependence, ANZIAM J., 58 (2016), 162-181.  doi: 10.1017/S144618111600016X. [27] S. E. A. Mohammed, Stochastic functional differential equations, Research Notes in Mathematics, 99. Pitman (Advanced Publishing Program), Boston, MA, 1984. [28] B. Øksendal and S. Agnès, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, (2001), 64–79. [29] Y. Shen and Y. Zeng, Optimal investment-reinsurance with delay for mean-variance insurers: A maximum principle approach, Insurance Math. Econom., 57 (2014), 1-12.  doi: 10.1016/j.insmatheco.2014.04.004. [30] H. M. Soner and W. H. Fleming, Controlled Markov Processes and Viscosity Solutions, 2$^nd$ edition, Springer-Verlag, New York, 2006. [31] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [32] K. C. Yuen, Z. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance Math. Econom., 64 (2015), 1-13.  doi: 10.1016/j.insmatheco.2015.04.009. [33] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Appl. Math. Optim., 42 (2000), 19-33.  doi: 10.1007/s002450010003.
The efficient frontiers for different $A$
The efficient frontiers for different $h$
The efficient frontiers for different $\beta$
The efficient frontiers for different $\lambda$
The efficient frontiers for different $\lambda_1$
The efficient frontiers for different $\lambda_2$
The efficient frontiers for different $\lambda_0$
The efficient frontiers for different $\sigma$
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