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doi: 10.3934/jimo.2022048
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Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model

Institute of Optics and Precision Mechanics, Ferhat Abbas University Setif 1, Setif 19000, Algeria

*Corresponding author: Ishak Alia

Received  July 2021 Revised  December 2021 Early access April 2022

This paper is devoted to study the open-loop equilibrium strategy for a mean-variance portfolio problem with investment constraints in a non-Markovian regime-switching jump-diffusion model. Specially, the investment strategies are constrained in a closed convex cone and all coefficients in the model are stochastic processes adapted to the filtration generated by a Markov chain. First, we provide a necessary and sufficient condition for an equilibrium strategy, which involves a system of forward and backward stochastic differential equations (FBSDEs, for short). Second, by solving these FBSDEs, we obtain a feedback representation of the equilibrium strategy. Third, we prove a theorem ensuring the almost everywhere uniqueness of the equilibrium solution. Finally, the results are applied to solve an example of the Markovian regime-switching model.

Citation: Ishak Alia, Mohamed Sofiane Alia. Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022048
References:
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I. Alia and F. Chighoub, Continuous-time mean-variance portfolio selection with regime-switching financial market: Time-consistent solution, Random Oper. Stoch. Equ., 29 (2021), 11-25.  doi: 10.1515/rose-2020-2050.

[2]

I. AliaF. Chighoub and A. Sohail, A characterization of equilibrium strategies in continuous-time mean-variance problems for insurers, Insurance Math. Econom., 68 (2016), 212-223.  doi: 10.1016/j.insmatheco.2016.03.009.

[3]

D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8.

[4]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016.  doi: 10.1093/rfs/hhq028.

[5]

I. Bajeux-Besnainou and R. Portait, Dynamic asset allocation in a mean-variance framework, Management Science, 44 (1998), 79-95.  doi: 10.1287/mnsc.44.11.S79.

[6]

A. BensoussanK. C. WongS. C. P. Yam and S. P. Yung, Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting, SIAM J. Financial Math., 5 (2014), 153-190.  doi: 10.1137/130914139.

[7]

T. R. Bielecki, H. Jin, S. Pliska and X. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Math. Finance, 15 (2005), 213–244, Available from: http://www.columbia.edu/xz2574/download/bjpz.pdf. doi: 10.1111/j.0960-1627.2005.00218.x.

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T. Björk and A. Murgoci, A general theory of Markovian time-inconsistent stochastic control problems, SSRN, (2010), Available from: https://ssrn.com/abstract=1694759.

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T. BjörkA. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.

[10]

P. Chen and H. Yang, Markowitz's mean-variance asset-liability management with regime switching: A multi period model, Appl. Math. Finance, 18 (2011), 29-50.  doi: 10.1080/13504861003703633.

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P. ChenH. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance Math. Econom., 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001.

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S. Crépey and A. Matoussi, Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison, Ann. Appl. Probab., 18 (2008), 2041-2069.  doi: 10.1214/08-AAP517.

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C. Czichowsky, Time-consistent mean-variance porftolio selection in discrete and continuous time, Finance Stoch., 17 (2013), 227-271.  doi: 10.1007/s00780-012-0189-9.

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M. DaiZ. Xu and X. Zhou, Continuous-time Markowitz's model with transaction costs, SIAM J. Financial Math., 1 (2010), 96-125.  doi: 10.1137/080742889.

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C. Donnelly, Sufficient stochastic maximum principle in the regime-switching diffusion model, Appl. Math. Optim., 64 (2011), 155-169.  doi: 10.1007/s00245-010-9130-9.

[19]

R. J. ElliottT. K. Siu and A. Badescu, On mean-variance portfolio selection under a hidden Markovian regime-switching model, Economic Modelling, 27 (2010), 678-686. 

[20]

J. M. Ingram and M. M. Marsh, Projections onto convex cones in Hilbert space, J. Approx. Theory, 64 (1991), 343-350.  doi: 10.1016/0021-9045(91)90067-K.

[21]

A. Jobert and L. C. G. Rogers, Option pricing with Markov-modulated dynamics, SIAM J. Control Optim., 44 (2006), 2063-2078.  doi: 10.1137/050623279.

[22]

Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear–quadratic control under constraint, preprint, arXiv: 1703.09415v1.

[23]

Y. HuH. Jin and X. Y. Zhou, Time-inconsistent stochastic linear quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.  doi: 10.1137/110853960.

[24]

Y. HuH. Jin and X. Y. Zhou, Time-inconsistent stochastic linear quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261-1279.  doi: 10.1137/15M1019040.

[25]

D. Li and W. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Math. Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.

[26]

A. E. B. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Math. Oper. Res., 29 (2004), 132-161.  doi: 10.1287/moor.1030.0065.

[27]

A. E. B. Lim and X. Zhou, Quadratic hedging and mean-variance portfolio selection with random parameters in a complete market, Math. Oper. Res., 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337.

[28]

H. M. Markowitz, Portfolio selection, Jornal of Finance, 7 (1952), 77–91, Available from: https://www.math.hkust.edu.hk/maykwok/courses/ma362/07F/markowitz_JF.pdf. doi: doi.org/10.2307/2975974.

[29] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9781107590120.
[30]

J. Sass and U. G. Haussmann, Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain, Finance Stoch., 8 (2004), 553-577.  doi: 10.1007/s00780-004-0132-9.

[31]

Y. ShenJ. Wei and Q. Zhao, Mean-variance asset-liability management problem under non-Markovian regime-switching model, Appl. Math. Optim., 81 (2020), 859-897.  doi: 10.1007/s00245-018-9523-8.

[32]

K. Si, Z. Xu, K. F. C. Yiu and X. Li, Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system, Journal of Industrial and Management Optimization. doi: 10.3934/jimo.2021074.

[33]

R. H. Stockbridge, Portfolio optimization in markets having stochastic rates, Stochastic Theory and Control, 280 (2002), 447-458.  doi: 10.1007/3-540-48022-6_30.

[34]

Z. Sun and X. Guo, Equilibrium for a time-inconsistent stochastic linear–quadratic control system with jumps and its application to the mean-variance problem, J. Optim. Theory Appl., 181 (2019), 383-410.  doi: 10.1007/s10957-018-01471-x.

[35]

H. SunZ. Sun and Y. Huang, Equilibrium investment and risk control for an insurer with non-Markovian regime-switching and no-shorting constraints, AIMS Math., 5 (2020), 6996-7013.  doi: 10.3934/math.2020449.

[36]

T. WangZ. Jin and J. Wei, Mean-variance portfolio selection under a non-Markovian regime-switching model: Time-consistent solutions, SIAM J. Control Optim., 57 (2019), 3249-3271.  doi: 10.1137/18M1186423.

[37]

T. Wang and J. Wei, Mean-variance portfolio selection under a non-Markovian regime switching model, J. Comput. Appl. Math., 350 (2019), 442-455.  doi: 10.1016/j.cam.2018.10.040.

[38]

J. Wei and T. Wang, Time-consistent mean-variance asset-liability management with random coefficients, Insurance Math. Econom., 77 (2017), 84-96.  doi: 10.1016/j.insmatheco.2017.08.011.

[39]

J. WeiK. C. WongS. C. P. Yam and S. P. Yung, Markowitz's mean–variance asset-liability management with regime switching: A time-consistent approach, Insurance Math. Econom., 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008.

[40]

J. Q. WenX. Li and J. Xiong, Weak closed-loop solvability of stochastic linear quadratic optimal control problems of Markovian regime switching system, Appl. Math. Optim., 84 (2021), 535-565.  doi: 10.1007/s00245-020-09653-8.

[41]

J. M. Xia, Mean-variance portfolio choice: Quadratic partial hedging, Math. Finance, 15 (2005), 533-538.  doi: 10.1111/j.1467-9965.2005.00231.x.

[42]

J. ZhangP. ChenZ. Jin and S. Li, Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model, Journal of Industrial and Management Optimization, 17 (2021), 765-777.  doi: 10.3934/jimo.2019133.

[43]

X. Zhang, X. Li and J. Xiong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems of Markovian regime switching system, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 69, 35 pp. doi: 10.1051/cocv/2021066.

[44]

Y. ZhangX. Li and S. Guo, Portfolio selection problems with Markowitz's mean–variance framework: A review of literature, Fuzzy Optim. Decis. Mak., 17 (2018), 125-158.  doi: 10.1007/s10700-017-9266-z.

[45]

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.

[46]

X. ZhangZ. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM J. Control Optim., 56 (2018), 2563-2592.  doi: 10.1137/17M112395X.

[47]

X. Y. Zhou and G. Yin, Markowitzs mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.

show all references

References:
[1]

I. Alia and F. Chighoub, Continuous-time mean-variance portfolio selection with regime-switching financial market: Time-consistent solution, Random Oper. Stoch. Equ., 29 (2021), 11-25.  doi: 10.1515/rose-2020-2050.

[2]

I. AliaF. Chighoub and A. Sohail, A characterization of equilibrium strategies in continuous-time mean-variance problems for insurers, Insurance Math. Econom., 68 (2016), 212-223.  doi: 10.1016/j.insmatheco.2016.03.009.

[3]

D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8.

[4]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016.  doi: 10.1093/rfs/hhq028.

[5]

I. Bajeux-Besnainou and R. Portait, Dynamic asset allocation in a mean-variance framework, Management Science, 44 (1998), 79-95.  doi: 10.1287/mnsc.44.11.S79.

[6]

A. BensoussanK. C. WongS. C. P. Yam and S. P. Yung, Time-consistent portfolio selection under short-selling prohibition: From discrete to continuous setting, SIAM J. Financial Math., 5 (2014), 153-190.  doi: 10.1137/130914139.

[7]

T. R. Bielecki, H. Jin, S. Pliska and X. Zhou, Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Math. Finance, 15 (2005), 213–244, Available from: http://www.columbia.edu/xz2574/download/bjpz.pdf. doi: 10.1111/j.0960-1627.2005.00218.x.

[8]

T. Björk and A. Murgoci, A general theory of Markovian time-inconsistent stochastic control problems, SSRN, (2010), Available from: https://ssrn.com/abstract=1694759.

[9]

T. BjörkA. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.

[10]

P. Chen and H. Yang, Markowitz's mean-variance asset-liability management with regime switching: A multi period model, Appl. Math. Finance, 18 (2011), 29-50.  doi: 10.1080/13504861003703633.

[11]

P. ChenH. Yang and G. Yin, Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model, Insurance Math. Econom., 43 (2008), 456-465.  doi: 10.1016/j.insmatheco.2008.09.001.

[12]

S. Cohen and R. Elliott, Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions, Ann. Appl. Probab., 20 (2010), 267-311.  doi: 10.1214/09-AAP619.

[13]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203485217.

[14]

S. Crepey, Financial Modeling: A Backward Stochastic Differential Equations Perspective Processes, Springer Finance. Springer Finance Textbooks. Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-37113-4.

[15]

S. Crépey and A. Matoussi, Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison, Ann. Appl. Probab., 18 (2008), 2041-2069.  doi: 10.1214/08-AAP517.

[16]

C. Czichowsky, Time-consistent mean-variance porftolio selection in discrete and continuous time, Finance Stoch., 17 (2013), 227-271.  doi: 10.1007/s00780-012-0189-9.

[17]

M. DaiZ. Xu and X. Zhou, Continuous-time Markowitz's model with transaction costs, SIAM J. Financial Math., 1 (2010), 96-125.  doi: 10.1137/080742889.

[18]

C. Donnelly, Sufficient stochastic maximum principle in the regime-switching diffusion model, Appl. Math. Optim., 64 (2011), 155-169.  doi: 10.1007/s00245-010-9130-9.

[19]

R. J. ElliottT. K. Siu and A. Badescu, On mean-variance portfolio selection under a hidden Markovian regime-switching model, Economic Modelling, 27 (2010), 678-686. 

[20]

J. M. Ingram and M. M. Marsh, Projections onto convex cones in Hilbert space, J. Approx. Theory, 64 (1991), 343-350.  doi: 10.1016/0021-9045(91)90067-K.

[21]

A. Jobert and L. C. G. Rogers, Option pricing with Markov-modulated dynamics, SIAM J. Control Optim., 44 (2006), 2063-2078.  doi: 10.1137/050623279.

[22]

Y. Hu, J. Huang and X. Li, Equilibrium for time-inconsistent stochastic linear–quadratic control under constraint, preprint, arXiv: 1703.09415v1.

[23]

Y. HuH. Jin and X. Y. Zhou, Time-inconsistent stochastic linear quadratic control, SIAM J. Control Optim., 50 (2012), 1548-1572.  doi: 10.1137/110853960.

[24]

Y. HuH. Jin and X. Y. Zhou, Time-inconsistent stochastic linear quadratic control: Characterization and uniqueness of equilibrium, SIAM J. Control Optim., 55 (2017), 1261-1279.  doi: 10.1137/15M1019040.

[25]

D. Li and W. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Math. Finance, 10 (2000), 387-406.  doi: 10.1111/1467-9965.00100.

[26]

A. E. B. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Math. Oper. Res., 29 (2004), 132-161.  doi: 10.1287/moor.1030.0065.

[27]

A. E. B. Lim and X. Zhou, Quadratic hedging and mean-variance portfolio selection with random parameters in a complete market, Math. Oper. Res., 27 (2002), 101-120.  doi: 10.1287/moor.27.1.101.337.

[28]

H. M. Markowitz, Portfolio selection, Jornal of Finance, 7 (1952), 77–91, Available from: https://www.math.hkust.edu.hk/maykwok/courses/ma362/07F/markowitz_JF.pdf. doi: doi.org/10.2307/2975974.

[29] L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, 2$^{nd}$ edition, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9781107590120.
[30]

J. Sass and U. G. Haussmann, Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain, Finance Stoch., 8 (2004), 553-577.  doi: 10.1007/s00780-004-0132-9.

[31]

Y. ShenJ. Wei and Q. Zhao, Mean-variance asset-liability management problem under non-Markovian regime-switching model, Appl. Math. Optim., 81 (2020), 859-897.  doi: 10.1007/s00245-018-9523-8.

[32]

K. Si, Z. Xu, K. F. C. Yiu and X. Li, Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system, Journal of Industrial and Management Optimization. doi: 10.3934/jimo.2021074.

[33]

R. H. Stockbridge, Portfolio optimization in markets having stochastic rates, Stochastic Theory and Control, 280 (2002), 447-458.  doi: 10.1007/3-540-48022-6_30.

[34]

Z. Sun and X. Guo, Equilibrium for a time-inconsistent stochastic linear–quadratic control system with jumps and its application to the mean-variance problem, J. Optim. Theory Appl., 181 (2019), 383-410.  doi: 10.1007/s10957-018-01471-x.

[35]

H. SunZ. Sun and Y. Huang, Equilibrium investment and risk control for an insurer with non-Markovian regime-switching and no-shorting constraints, AIMS Math., 5 (2020), 6996-7013.  doi: 10.3934/math.2020449.

[36]

T. WangZ. Jin and J. Wei, Mean-variance portfolio selection under a non-Markovian regime-switching model: Time-consistent solutions, SIAM J. Control Optim., 57 (2019), 3249-3271.  doi: 10.1137/18M1186423.

[37]

T. Wang and J. Wei, Mean-variance portfolio selection under a non-Markovian regime switching model, J. Comput. Appl. Math., 350 (2019), 442-455.  doi: 10.1016/j.cam.2018.10.040.

[38]

J. Wei and T. Wang, Time-consistent mean-variance asset-liability management with random coefficients, Insurance Math. Econom., 77 (2017), 84-96.  doi: 10.1016/j.insmatheco.2017.08.011.

[39]

J. WeiK. C. WongS. C. P. Yam and S. P. Yung, Markowitz's mean–variance asset-liability management with regime switching: A time-consistent approach, Insurance Math. Econom., 53 (2013), 281-291.  doi: 10.1016/j.insmatheco.2013.05.008.

[40]

J. Q. WenX. Li and J. Xiong, Weak closed-loop solvability of stochastic linear quadratic optimal control problems of Markovian regime switching system, Appl. Math. Optim., 84 (2021), 535-565.  doi: 10.1007/s00245-020-09653-8.

[41]

J. M. Xia, Mean-variance portfolio choice: Quadratic partial hedging, Math. Finance, 15 (2005), 533-538.  doi: 10.1111/j.1467-9965.2005.00231.x.

[42]

J. ZhangP. ChenZ. Jin and S. Li, Open-loop equilibrium strategy for mean-variance portfolio selection: A log-return model, Journal of Industrial and Management Optimization, 17 (2021), 765-777.  doi: 10.3934/jimo.2019133.

[43]

X. Zhang, X. Li and J. Xiong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems of Markovian regime switching system, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 69, 35 pp. doi: 10.1051/cocv/2021066.

[44]

Y. ZhangX. Li and S. Guo, Portfolio selection problems with Markowitz's mean–variance framework: A review of literature, Fuzzy Optim. Decis. Mak., 17 (2018), 125-158.  doi: 10.1007/s10700-017-9266-z.

[45]

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.

[46]

X. ZhangZ. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM J. Control Optim., 56 (2018), 2563-2592.  doi: 10.1137/17M112395X.

[47]

X. Y. Zhou and G. Yin, Markowitzs mean–variance portfolio selection with regime switching: A continuous-time model, SIAM J. Control Optim., 42 (2003), 1466-1482.  doi: 10.1137/S0363012902405583.

Figure 1.  The investment ratio in the two regimes with $ \mu = 1.2 $
Figure 2.  The impact of the parameter $ \mu $ on the investment ratio in regime 1
Figure 3.  The impact of the parameter $ \mu $ on the investment ratio in regime 2
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