# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018166

## Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching

 1 Department of Economics, The University of Melbourne, VIC 3010, Australia 2 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China

* Corresponding author: Haixiang Yao. Tel.:+86 2037105360

Received  January 2018 Revised  April 2018 Published  October 2018

Fund Project: This research was supported by the National Natural Science Foundation of China (Nos. 71871071, 71471045, 71721001), the Natural Science Foundation of Guangdong Province of China (Nos. 2018B030311004, 2017A030313399, 2017A030313397), the Innovation Team Project of Guangdong Colleges and Universities (No. 2016WCXTD012), the Innovative School Project in Higher Education of Guangdong Province of China (No. GWTP-GC-2017-03)

The present article investigates a continuous-time mean-variance portfolio selection problem with regime-switching under the constraint of no-shorting. The literature along this line is essentially dominated by the Hamilton-Jacobi-Bellman (HJB) equation approach. However, in the presence of switching regimes, a system of HJB equations rather than a single equation need to be tackled concurrently, which might not be solvable in terms of classical solutions, or even not in the weaker viscosity sense as well. Instead, we first introduce a general result on the sign of geometric Brownian motion with jumps, then derive the efficient portfolio and frontier via the maximum principle approach; in particular, we observe, under a mild technical assumption on the initial conditions, that the no-shorting constraint will consistently be satisfied over the whole finite time horizon. Further numerical illustrations will be provided.

Citation: Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018166
##### References:
 [1] L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205. doi: 10.1007/s00186-007-0195-4. [2] A. Bensoussan, K. J. Sung, S. C. H. Yam and S. P. Yung, A non-zero stochastic differential reinsurance game with mixed regime-switching, working paper, 2011. [3] J. N. Bi, J. Y. Guo and L. H. Bai, Optimal multi-asset investment with no-shorting constraint under mean-variance criterion for an insurer, Journal of Systems Science and Complexity, 24 (2011), 291-307. doi: 10.1007/s11424-011-8014-7. [4] T. Björk, Finite dimensional optimal filters for a class of Itô-processes with jumping parameters, Stochastics, 4 (1980), 167-183. doi: 10.1080/17442508008833160. [5] E. Çanakoǧlu and S. Özekici, Portfolio selection in stochastic markets with HARA utility functions, European Journal of Operational Research, 201 (2010), 520-536. doi: 10.1016/j.ejor.2009.03.017. [6] 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. [7] Y. Z. Hu, Multi-dimensional geometric Brownian motions, Onsager-Machlup functions, and applications to mathematical finance, Acta Mathematica Scientia, 20 (2000), 341-358. doi: 10.1016/S0252-9602(17)30641-0. [8] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100. [9] X. Li and Z. Q. Xu, Continuous-time Markowitzs model with constraints on wealth and portfolio, Operations Research Letters, 44 (2016), 729-736. doi: 10.1016/j.orl.2016.09.004. [10] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shoring constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555. doi: 10.1137/S0363012900378504. [11] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [12] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous - time case, Review of Economics and Statistics, 51 (1969), 247-257. [13] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. [14] P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 51 (1969), 239-246. [15] L. R. Sotomayor and A. Cadenillas, Explicit solutions of consumption-investment problems in financial markets with regime switching, Mathematical Finance, 19 (2009), 251-279. doi: 10.1111/j.1467-9965.2009.00366.x. [16] 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. [17] H. Wu and H. Chen, Nash equilibrium strategy for a multi-period mean-ariance portfolio selection problem with regime switching, Economic Modelling, 46 (2015), 79-90. [18] G. L. Xu and S. E. Shreve, A duality method for optimal consumption and investment under short-selling prohibition: Ⅱ. Constant market coefficients, The Annals of Applied Probability, 2 (1992), 314-328. doi: 10.1214/aoap/1177005706. [19] M. Zhang and P. Chen, Mean-variance portfolio selection with regime switching under shorting prohibition, Operations Research Letters, 44 (2016), 658-662. doi: 10.1016/j.orl.2016.07.008. [20] 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.

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##### References:
 [1] L. H. Bai and H. Y. Zhang, Dynamic mean-variance problem with constrained risk control for the insurers, Mathematical Methods of Operations Research, 68 (2008), 181-205. doi: 10.1007/s00186-007-0195-4. [2] A. Bensoussan, K. J. Sung, S. C. H. Yam and S. P. Yung, A non-zero stochastic differential reinsurance game with mixed regime-switching, working paper, 2011. [3] J. N. Bi, J. Y. Guo and L. H. Bai, Optimal multi-asset investment with no-shorting constraint under mean-variance criterion for an insurer, Journal of Systems Science and Complexity, 24 (2011), 291-307. doi: 10.1007/s11424-011-8014-7. [4] T. Björk, Finite dimensional optimal filters for a class of Itô-processes with jumping parameters, Stochastics, 4 (1980), 167-183. doi: 10.1080/17442508008833160. [5] E. Çanakoǧlu and S. Özekici, Portfolio selection in stochastic markets with HARA utility functions, European Journal of Operational Research, 201 (2010), 520-536. doi: 10.1016/j.ejor.2009.03.017. [6] 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. [7] Y. Z. Hu, Multi-dimensional geometric Brownian motions, Onsager-Machlup functions, and applications to mathematical finance, Acta Mathematica Scientia, 20 (2000), 341-358. doi: 10.1016/S0252-9602(17)30641-0. [8] D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100. [9] X. Li and Z. Q. Xu, Continuous-time Markowitzs model with constraints on wealth and portfolio, Operations Research Letters, 44 (2016), 729-736. doi: 10.1016/j.orl.2016.09.004. [10] X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shoring constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555. doi: 10.1137/S0363012900378504. [11] H. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. [12] R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous - time case, Review of Economics and Statistics, 51 (1969), 247-257. [13] R. C. Merton, Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3 (1971), 373-413. doi: 10.1016/0022-0531(71)90038-X. [14] P. A. Samuelson, Lifetime portfolio selection by dynamic stochastic programming, The Review of Economics and Statistics, 51 (1969), 239-246. [15] L. R. Sotomayor and A. Cadenillas, Explicit solutions of consumption-investment problems in financial markets with regime switching, Mathematical Finance, 19 (2009), 251-279. doi: 10.1111/j.1467-9965.2009.00366.x. [16] 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. [17] H. Wu and H. Chen, Nash equilibrium strategy for a multi-period mean-ariance portfolio selection problem with regime switching, Economic Modelling, 46 (2015), 79-90. [18] G. L. Xu and S. E. Shreve, A duality method for optimal consumption and investment under short-selling prohibition: Ⅱ. Constant market coefficients, The Annals of Applied Probability, 2 (1992), 314-328. doi: 10.1214/aoap/1177005706. [19] M. Zhang and P. Chen, Mean-variance portfolio selection with regime switching under shorting prohibition, Operations Research Letters, 44 (2016), 658-662. doi: 10.1016/j.orl.2016.07.008. [20] 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 value of the stochastic process $P(t, \alpha(t))[x(t)+(\lambda-z)H(t, \alpha(t))]$
A sample path of the efficient portfolio $u^{*} = (u_1, u_2, u_3)'$
The process $-[x(t)+(\lambda^{*}-z)H(t, \alpha(t))]$
The corresponding efficient frontier
The effects of initial market mode $i_0$
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