# American Institute of Mathematical Sciences

January  2022, 18(1): 341-366. doi: 10.3934/jimo.2020156

## Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach

 1 School of Finance, Nanjing University of Finance and Economics, Nanjing 210023, China 2 School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China 3 Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong

* Corresponding author: Zhibin Liang

Received  February 2019 Revised  June 2020 Published  January 2022 Early access  November 2020

Fund Project: This research is supported by National Natural Science Foundation of China (Grant No.12071224 and Grant No.11771079), the Research Grants Council of the Hong Kong Special Administrative Region (Project No. HKU17306220), the Philosophy and Social Science Foundation for Colleges and Universities in Jiangsu Province (Grant No.2020SJA0261), and the MOE Project of Humanities and Social Sciences (Grant No.19YJCZH083)

In this paper, an optimal portfolio selection problem with mean-variance utility is considered for a financial market consisting of one risk-free asset and two risky assets, whose price processes are modulated by jump-diffusion model, the two jump number processes are correlated through a common shock, and the Brownian motions are supposed to be dependent. Moreover, it is assumed that not only the risk aversion coefficient but also the market parameters such as the appreciation and volatility rates as well as the jump amplitude depend on a Markov chain with finite states. In addition, short selling is supposed to be prohibited. Using the technique of stochastic control theory and the corresponding extended Hamilton-Jacobi-Bellman equation, the explicit expressions of the optimal strategies and value function are obtained within a game theoretic framework, and the existence and uniqueness of the solutions are proved as well. In the end, some numerical examples are presented to show the impact of the parameters on the optimal strategies, and some further discussions on the case of $n\geq 3$ risky assets are given to demonstrate the important effect of the correlation coefficient of the Brownian motions on the optimal results.

Citation: Caibin Zhang, Zhibin Liang, Kam Chuen Yuen. Portfolio optimization for jump-diffusion risky assets with regime switching: A time-consistent approach. Journal of Industrial and Management Optimization, 2022, 18 (1) : 341-366. doi: 10.3934/jimo.2020156
##### References:
 [1] N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1. [2] A. Bensoussan, K. C. Wong, S. C. P. Yam and S. P. Yung, Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting, SIAM Journal on Financial Mathematics, 5 (2014), 153-190.  doi: 10.1137/130914139. [3] A. Bensoussan, K. C. Wong and S. C. P. Yam, A paradox in time-consistency in the mean-variance problem, Finance and Stochastics, 23 (2019), 173-207.  doi: 10.1007/s00780-018-00381-0. [4] 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. [5] J. Bi and J. Guo, Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, Journal of Optimization Theory and Applications, 157 (2013), 252-275.  doi: 10.1007/s10957-012-0138-y. [6] J. Bi, Z. Liang and F. Xu, Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence, Insurance: Mathematics and Economics, 70 (2016), 245-258.  doi: 10.1016/j.insmatheco.2016.06.012. [7] T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, Working Paper, Stockholm School of Economics, Sweden, 2010. [8] T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x. [9] P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001. [10] R. J. Elliott and J. Hoek, An application of hidden Markov models to asset allocation problems, Finance and Stochastics, 1 (1997), 229-238. [11] P. A. Forsyth and J. Wang, Continuous time mean variance asset allocation: A time-consistent strategy, European Journal of Operational Research, 209 (2011), 184-201.  doi: 10.1016/j.ejor.2010.09.038. [12] D. Li and W. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [13] Z. Liang and M. Song, Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information, Insurance: Mathematics and Economics, 65 (2015), 66-76.  doi: 10.1016/j.insmatheco.2015.08.008. [14] Z. Liang, J. Bi, K. C. Yuen and C. Zhang, Optimal mean-variance reinsurance and investment in a jump-diffusion financial market with common shock dependence, Mathematical Methods of Operations Research, 84 (2016), 155-181.  doi: 10.1007/s00186-016-0538-0. [15] Z. Liang and K. C. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scandinavian Actuarial Journal, 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899. [16] Z. Liang, K. C. Yuen and C. Zhang, Optimal reinsurance and investment in a jump-diffusion financial market with common shock dependence, Journal of Applied Mathematics and Computing, 56 (2018), 637-664.  doi: 10.1007/s12190-017-1119-y. [17] A. E. B. Lim and X. Y. Zhou, Quadratic hedging and mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337. [18] H. M. Markowitz, Portfolio Selection, John Wiley & Sons, Inc., New York, Chapman & Hall, Ltd., London, 1959. [19] Z. Ming, Z. Liang and C. Zhang, Optimal mean-variance reinsurance with dependent risks, ANZIAM Journal, 58 (2016), 162-181.  doi: 10.1017/S144618111600016X. [20] J. L. Pedersen and G. Peskir, Optimal mean-variance portfolio selection, Mathematics and Financial Economics, 11 (2017), 137-160.  doi: 10.1007/s11579-016-0174-8. [21] H. R. Richardson, A minimum variance result in continuous trading portfolio optimization, Management Science, 35 (1989), 1045-1055.  doi: 10.1287/mnsc.35.9.1045. [22] Z. Sun and J. Guo, Optimal mean-variance investment and reinsurance problem for an insurer with stochastic volatility, Mathematical Methods of Operations Research, 88 (2018), 59-79.  doi: 10.1007/s00186-017-0628-7. [23] J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Markowitz's mean-variance asset-liability management with regime switching: A time-consistent approach, Insurance: Mathematics and Economics, 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008. [24] J. Wei, D. Li and Y. Zeng, Robust optimal consumption-investment strategy with nonexponential discounting, Journal of Industrial and Management Optimization, 16 (2020), 207-230.  doi: 10.3934/jimo.2018147. [25] J. Wei, H. Yang and R. Wang, Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching, Journal of Optimization Theory and Applications, 147 (2010), 358-377.  doi: 10.1007/s10957-010-9726-x. [26] K. C. Yuen, Z. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance: Mathematics and Economics, 64 (2015), 1-13.  doi: 10.1016/j.insmatheco.2015.04.009. [27] Y. Zeng, Z. Li and Y. Lai, Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps, Insurance: Mathematics and Economics, 52 (2013), 498-507.  doi: 10.1016/j.insmatheco.2013.02.007. [28] C. Zhang and Z. Liang, Portfolio optimization for jump-diffusion risky assets with common shock dependence and state dependent risk aversion, Optimal Control Applications and Methods, 38 (2017), 229-246.  doi: 10.1002/oca.2252. [29] X. Zhang and T. K. Siu, On optimal proportional reinsurance and investment in a Markovian regime-switching economy, Acta Mathematica Sinica English Series, 28 (2012), 67-82.  doi: 10.1007/s10114-012-9761-7. [30] J. Zhou, X. Yang and J. Guo, Portfolio selection and risk control for an insurer in the Lévy market under mean-variance criterion, Statistics and Probability Letters, 126 (2017), 139-149.  doi: 10.1016/j.spl.2017.03.008. [31] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003. [32] X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.

show all references

##### References:
 [1] N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1. [2] A. Bensoussan, K. C. Wong, S. C. P. Yam and S. P. Yung, Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting, SIAM Journal on Financial Mathematics, 5 (2014), 153-190.  doi: 10.1137/130914139. [3] A. Bensoussan, K. C. Wong and S. C. P. Yam, A paradox in time-consistency in the mean-variance problem, Finance and Stochastics, 23 (2019), 173-207.  doi: 10.1007/s00780-018-00381-0. [4] 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. [5] J. Bi and J. Guo, Optimal mean-variance problem with constrained controls in a jump-diffusion financial market for an insurer, Journal of Optimization Theory and Applications, 157 (2013), 252-275.  doi: 10.1007/s10957-012-0138-y. [6] J. Bi, Z. Liang and F. Xu, Optimal mean-variance investment and reinsurance problems for the risk model with common shock dependence, Insurance: Mathematics and Economics, 70 (2016), 245-258.  doi: 10.1016/j.insmatheco.2016.06.012. [7] T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, Working Paper, Stockholm School of Economics, Sweden, 2010. [8] T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state dependent risk aversion, Mathematical Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x. [9] P. Chen, H. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance: Mathematics and Economics, 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001. [10] R. J. Elliott and J. Hoek, An application of hidden Markov models to asset allocation problems, Finance and Stochastics, 1 (1997), 229-238. [11] P. A. Forsyth and J. Wang, Continuous time mean variance asset allocation: A time-consistent strategy, European Journal of Operational Research, 209 (2011), 184-201.  doi: 10.1016/j.ejor.2010.09.038. [12] D. Li and W. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100. [13] Z. Liang and M. Song, Time-consistent reinsurance and investment strategies for mean-variance insurer under partial information, Insurance: Mathematics and Economics, 65 (2015), 66-76.  doi: 10.1016/j.insmatheco.2015.08.008. [14] Z. Liang, J. Bi, K. C. Yuen and C. Zhang, Optimal mean-variance reinsurance and investment in a jump-diffusion financial market with common shock dependence, Mathematical Methods of Operations Research, 84 (2016), 155-181.  doi: 10.1007/s00186-016-0538-0. [15] Z. Liang and K. C. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scandinavian Actuarial Journal, 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899. [16] Z. Liang, K. C. Yuen and C. Zhang, Optimal reinsurance and investment in a jump-diffusion financial market with common shock dependence, Journal of Applied Mathematics and Computing, 56 (2018), 637-664.  doi: 10.1007/s12190-017-1119-y. [17] A. E. B. Lim and X. Y. Zhou, Quadratic hedging and mean-variance portfolio selection with random parameters in a complete market, Mathematics of Operations Research, 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337. [18] H. M. Markowitz, Portfolio Selection, John Wiley & Sons, Inc., New York, Chapman & Hall, Ltd., London, 1959. [19] Z. Ming, Z. Liang and C. Zhang, Optimal mean-variance reinsurance with dependent risks, ANZIAM Journal, 58 (2016), 162-181.  doi: 10.1017/S144618111600016X. [20] J. L. Pedersen and G. Peskir, Optimal mean-variance portfolio selection, Mathematics and Financial Economics, 11 (2017), 137-160.  doi: 10.1007/s11579-016-0174-8. [21] H. R. Richardson, A minimum variance result in continuous trading portfolio optimization, Management Science, 35 (1989), 1045-1055.  doi: 10.1287/mnsc.35.9.1045. [22] Z. Sun and J. Guo, Optimal mean-variance investment and reinsurance problem for an insurer with stochastic volatility, Mathematical Methods of Operations Research, 88 (2018), 59-79.  doi: 10.1007/s00186-017-0628-7. [23] J. Wei, K. C. Wong, S. C. P. Yam and S. P. Yung, Markowitz's mean-variance asset-liability management with regime switching: A time-consistent approach, Insurance: Mathematics and Economics, 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008. [24] J. Wei, D. Li and Y. Zeng, Robust optimal consumption-investment strategy with nonexponential discounting, Journal of Industrial and Management Optimization, 16 (2020), 207-230.  doi: 10.3934/jimo.2018147. [25] J. Wei, H. Yang and R. Wang, Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching, Journal of Optimization Theory and Applications, 147 (2010), 358-377.  doi: 10.1007/s10957-010-9726-x. [26] K. C. Yuen, Z. Liang and M. Zhou, Optimal proportional reinsurance with common shock dependence, Insurance: Mathematics and Economics, 64 (2015), 1-13.  doi: 10.1016/j.insmatheco.2015.04.009. [27] Y. Zeng, Z. Li and Y. Lai, Time-consistent investment and reinsurance strategies for mean-variance insurers with jumps, Insurance: Mathematics and Economics, 52 (2013), 498-507.  doi: 10.1016/j.insmatheco.2013.02.007. [28] C. Zhang and Z. Liang, Portfolio optimization for jump-diffusion risky assets with common shock dependence and state dependent risk aversion, Optimal Control Applications and Methods, 38 (2017), 229-246.  doi: 10.1002/oca.2252. [29] X. Zhang and T. K. Siu, On optimal proportional reinsurance and investment in a Markovian regime-switching economy, Acta Mathematica Sinica English Series, 28 (2012), 67-82.  doi: 10.1007/s10114-012-9761-7. [30] J. Zhou, X. Yang and J. Guo, Portfolio selection and risk control for an insurer in the Lévy market under mean-variance criterion, Statistics and Probability Letters, 126 (2017), 139-149.  doi: 10.1016/j.spl.2017.03.008. [31] X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33.  doi: 10.1007/s002450010003. [32] X. Y. Zhou and G. Yin, Markowitz's mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization, 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.
The comparison of value functions with and without constraint
The comparison of optimal strategies between regime 1 and regime 2
The effect of parameter $\lambda_{01}$ on optimal strategies with regime 1
The effect of jump's parameters on optimal strategies with regime 1
The effect of risk-free rate on optimal strategies with regime 1
The value of parameters
 parameters $T$ $x$ $\gamma_{i}$ $r_0$ $r_{1i}$ $r_{2i}$ $\rho$ $\sigma_{1i}$ $\sigma_{2i}$ $\mu_{1i}$ $\mu_{2i}$ $\beta_{1i}$ $\beta_{2i}$ $\lambda_{1i}$ $\lambda_{2i}$ $\lambda_{0i}$ $e_1$(bullish) 5 2 0.5 0.3 0.60 0.30 0.2 0.3 0.8 0.06 0.08 0.02 0.055 2 0.5 1 $e_2$(bearish) 5 2 1 0.3 0.25 0.07 0.1 0.4 0.5 0.03 0.04 0.03 0.032 1 0.3 0.6
 parameters $T$ $x$ $\gamma_{i}$ $r_0$ $r_{1i}$ $r_{2i}$ $\rho$ $\sigma_{1i}$ $\sigma_{2i}$ $\mu_{1i}$ $\mu_{2i}$ $\beta_{1i}$ $\beta_{2i}$ $\lambda_{1i}$ $\lambda_{2i}$ $\lambda_{0i}$ $e_1$(bullish) 5 2 0.5 0.3 0.60 0.30 0.2 0.3 0.8 0.06 0.08 0.02 0.055 2 0.5 1 $e_2$(bearish) 5 2 1 0.3 0.25 0.07 0.1 0.4 0.5 0.03 0.04 0.03 0.032 1 0.3 0.6
The parameters-set
 parameters $T$ $x$ $\gamma_{i}$ $r_0$ $r_{1i}$ $r_{2i}$ $\rho$ $\sigma_{1i}$ $\sigma_{2i}$ $\mu_{1i}$ $\mu_{2i}$ $\beta_{1i}$ $\beta_{2i}$ $\lambda_{1i}$ $\lambda_{2i}$ $\lambda_{0i}$ $e_1$(bullish) 5 2 0.5 0.3 0.60 0.65 0.2 0.7 0.8 0.06 0.08 0.04 0.045 2 0.5 1 $e_2$(bearish) 5 2 1 0.3 0.35 0.37 0.1 0.4 0.5 0.03 0.04 0.03 0.032 1 0.3 0.6
 parameters $T$ $x$ $\gamma_{i}$ $r_0$ $r_{1i}$ $r_{2i}$ $\rho$ $\sigma_{1i}$ $\sigma_{2i}$ $\mu_{1i}$ $\mu_{2i}$ $\beta_{1i}$ $\beta_{2i}$ $\lambda_{1i}$ $\lambda_{2i}$ $\lambda_{0i}$ $e_1$(bullish) 5 2 0.5 0.3 0.60 0.65 0.2 0.7 0.8 0.06 0.08 0.04 0.045 2 0.5 1 $e_2$(bearish) 5 2 1 0.3 0.35 0.37 0.1 0.4 0.5 0.03 0.04 0.03 0.032 1 0.3 0.6
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