# American Institute of Mathematical Sciences

July  2022, 18(4): 2415-2433. doi: 10.3934/jimo.2021074

## Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system

 1 School of Mathematics, Shandong University, Jinan, Shandong Province, 250100, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

* Corresponding author: Ka Fai Cedric Yiu

Received  September 2020 Revised  January 2021 Published  July 2022 Early access  April 2021

This paper investigates the mean-field stochastic linear quadratic optimal control problem of Markov regime switching system (M-MF-SLQ, for short). The representation of the cost functional for the M-MF-SLQ is derived using the technique of operators. It is shown that the convexity of the cost functional is necessary for the finiteness of the M-MF-SLQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex cost functionals, a characterization of the finiteness of the problem is derived and a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. We demonstrate with a few examples that our results can be employed for tackling some financial problems such as mean-variance portfolio selection problem.

Citation: Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2415-2433. doi: 10.3934/jimo.2021074
##### References:
 [1] D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Applied Mathematics and Optimization, 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8. [2] J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM Journal on Control and Optimization, 14 (1976), 419–444. doi: 10.1137/0314028. [3] R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Applied Mathematics and Optimization, 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y. [4] R. Buckdahn, B. Djehiche, J. Li and S. Peng, Mean-field backward stochastic differential equations: A limit approach, Annals of Probability, 37 (2009), 1524-1565.  doi: 10.1214/08-AOP442. [5] X. Cui, X. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Transactions on Automatic Control, 59 (2014), 1833–1844. doi: 10.1109/TAC.2014.2311875. [6] C. Donnelly, Suffcient stochastic maximum principle in a regime-switching diffusion model, Applied Mathematics & Optimization, 64 (2011), 155-169.  doi: 10.1007/s00245-010-9130-9. [7] C. Donnelly and A. J. Heunis, Quadratic Risk Minimization in a Regime-Switching Model with Portfolio Constraints, SIAM Journal on Control and Optimization, 50 (2012), 2431–2461. doi: 10.1137/100809271. [8] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Applications of Mathematics, New York: Springer-Verlag, 1995. [9] R. Elliott, X. Li and Y.-H. Ni, Discrete time mean-field stochastic linear-quadratic optimal control problems, Automatica, 49 (2013), 3222-3233.  doi: 10.1016/j.automatica.2013.08.017. [10] J. Huang, X. Li and J. Yong, A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon, Mathematical Control & Related Fields, 5 (2015), 97-139.  doi: 10.3934/mcrf.2015.5.97. [11] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197. [12] H. Kushner, Optimal stochastic control, IRE Transactions on Automatic Control, 7 (1962), 120–122. doi: 10.1109/TAC.1962.1105490. [13] X. Li and X. Y. Zhou, Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon, Communications in Information and Systems, 2 (2002), 265-282.  doi: 10.4310/CIS.2002.v2.n3.a4. [14] X. Li, J. Sun and J. Yong, Mean-field stochastic linear quadratic optimal control problems: Closed-loop solvability, Probability, Uncertainty and Quantitative Risk, 1 (2016), Paper No. 2, 24 pp. doi: 10.1186/s41546-016-0002-3. [15] X. Li, X. Y. Zhou and M. Ait Rami, Indefinite stochastic linear quadratic control with Markovian jumps in infinite time horizon, Journal of Global Optimization, 27 (2003), 149-175.  doi: 10.1023/A:1024887007165. [16] X. Li and X. Y. Zhou, Continuous-time mean-variance efficiency: The 80% rule, The Annals of Applied Probability, 16 (2006), 1751-1763.  doi: 10.1214/105051606000000349. [17] Y. Liu, G. Yin and X. Y. Zhou, Near-optimal controls of random-switching LQ problems with indefinite control weight costs, Automatica, 41 (2005), 1063-1070.  doi: 10.1016/j.automatica.2005.01.002. [18] R. Penrose, A generalized inverse for matrices, Mathematical Proceedings of the Cambridge Philosophical Society, 51 (1955), 406-413.  doi: 10.1017/S0305004100030401. [19] J. Sun and J. Yong, Linear quadratic stochastic differential games: open-loop and closed-loop saddle points, SIAM Journal on Control and Optimization, 52 (2014), 4082-4121.  doi: 10.1137/140953642. [20] J. Sun, X. Li and J. Yong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems, SIAM Journal on Control and Optimization, 54 (2016), 2274–2308. doi: 10.1137/15M103532X. [21] J. Sun, Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Open-Loop Solvabilities, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 1099-1127.  doi: 10.1051/cocv/2016023. [22] R. Tao and Z. Wu, Maximum principle for optimal control problems of forward-backward regime-switching system and applications, Systems & Control Letters, 61 (2012), 911-917.  doi: 10.1016/j.sysconle.2012.06.006. [23] W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM Journal on Control, 6 (1968), 681–697. doi: 10.1137/0306044. [24] Z. Wu and X.-R. Wang, FBSDE with Poisson process and its application to linear quadratic stochastic optimal control problem with random jumps, Acta Automatica Sinica, 29 (2003), 821-826. [25] G. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd Edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4346-9. [26] K.-F. C. Yiu, J. Liu, T. K. Siu and W.-K. Ching, Optimal portfolios with regime switching and value-at-risk constraint, Automatica, 46 (2010), 979-989.  doi: 10.1016/j.automatica.2010.02.027. [27] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer New York, 1999. doi: 10.1007/978-1-4612-1466-3. [28] J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM Journal on Control and Optimization, 51 (2013), 2809-2838.  doi: 10.1137/120892477. [29] Z. Yu, Infinite horizon jump-diffusion forward-backward stochastic differential equations and their application to backward linear-quadratic problems, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 1331–1359. doi: 10.1051/cocv/2016055. [30] X. Zhang, R. J. Elliott, T. K. Siu and J. Guo, Markovian regime-switching market completion using additional markov jump assets, IMA Journal of Management Mathematics, 23 (2012), 283–305. doi: 10.1093/imaman/dpr018. [31] X. Zhang and X. Li, Open-Loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems of Markov regime-switching system, arXiv: 1809.01891, 2018. [32] X. Zhang, Z. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM Journal on Control and Optimization, 56 (2018), 2563–2592. doi: 10.1137/17M112395X. [33] 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] D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Applied Mathematics and Optimization, 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8. [2] J.-M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM Journal on Control and Optimization, 14 (1976), 419–444. doi: 10.1137/0314028. [3] R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Applied Mathematics and Optimization, 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y. [4] R. Buckdahn, B. Djehiche, J. Li and S. Peng, Mean-field backward stochastic differential equations: A limit approach, Annals of Probability, 37 (2009), 1524-1565.  doi: 10.1214/08-AOP442. [5] X. Cui, X. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Transactions on Automatic Control, 59 (2014), 1833–1844. doi: 10.1109/TAC.2014.2311875. [6] C. Donnelly, Suffcient stochastic maximum principle in a regime-switching diffusion model, Applied Mathematics & Optimization, 64 (2011), 155-169.  doi: 10.1007/s00245-010-9130-9. [7] C. Donnelly and A. J. Heunis, Quadratic Risk Minimization in a Regime-Switching Model with Portfolio Constraints, SIAM Journal on Control and Optimization, 50 (2012), 2431–2461. doi: 10.1137/100809271. [8] R. J. Elliott, L. Aggoun and J. B. Moore, Hidden Markov Models: Estimation and Control, Applications of Mathematics, New York: Springer-Verlag, 1995. [9] R. Elliott, X. Li and Y.-H. Ni, Discrete time mean-field stochastic linear-quadratic optimal control problems, Automatica, 49 (2013), 3222-3233.  doi: 10.1016/j.automatica.2013.08.017. [10] J. Huang, X. Li and J. Yong, A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon, Mathematical Control & Related Fields, 5 (2015), 97-139.  doi: 10.3934/mcrf.2015.5.97. [11] M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197. [12] H. Kushner, Optimal stochastic control, IRE Transactions on Automatic Control, 7 (1962), 120–122. doi: 10.1109/TAC.1962.1105490. [13] X. Li and X. Y. Zhou, Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon, Communications in Information and Systems, 2 (2002), 265-282.  doi: 10.4310/CIS.2002.v2.n3.a4. [14] X. Li, J. Sun and J. Yong, Mean-field stochastic linear quadratic optimal control problems: Closed-loop solvability, Probability, Uncertainty and Quantitative Risk, 1 (2016), Paper No. 2, 24 pp. doi: 10.1186/s41546-016-0002-3. [15] X. Li, X. Y. Zhou and M. Ait Rami, Indefinite stochastic linear quadratic control with Markovian jumps in infinite time horizon, Journal of Global Optimization, 27 (2003), 149-175.  doi: 10.1023/A:1024887007165. [16] X. Li and X. Y. Zhou, Continuous-time mean-variance efficiency: The 80% rule, The Annals of Applied Probability, 16 (2006), 1751-1763.  doi: 10.1214/105051606000000349. [17] Y. Liu, G. Yin and X. Y. Zhou, Near-optimal controls of random-switching LQ problems with indefinite control weight costs, Automatica, 41 (2005), 1063-1070.  doi: 10.1016/j.automatica.2005.01.002. [18] R. Penrose, A generalized inverse for matrices, Mathematical Proceedings of the Cambridge Philosophical Society, 51 (1955), 406-413.  doi: 10.1017/S0305004100030401. [19] J. Sun and J. Yong, Linear quadratic stochastic differential games: open-loop and closed-loop saddle points, SIAM Journal on Control and Optimization, 52 (2014), 4082-4121.  doi: 10.1137/140953642. [20] J. Sun, X. Li and J. Yong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems, SIAM Journal on Control and Optimization, 54 (2016), 2274–2308. doi: 10.1137/15M103532X. [21] J. Sun, Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Open-Loop Solvabilities, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 1099-1127.  doi: 10.1051/cocv/2016023. [22] R. Tao and Z. Wu, Maximum principle for optimal control problems of forward-backward regime-switching system and applications, Systems & Control Letters, 61 (2012), 911-917.  doi: 10.1016/j.sysconle.2012.06.006. [23] W. M. Wonham, On a matrix Riccati equation of stochastic control, SIAM Journal on Control, 6 (1968), 681–697. doi: 10.1137/0306044. [24] Z. Wu and X.-R. Wang, FBSDE with Poisson process and its application to linear quadratic stochastic optimal control problem with random jumps, Acta Automatica Sinica, 29 (2003), 821-826. [25] G. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd Edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4346-9. [26] K.-F. C. Yiu, J. Liu, T. K. Siu and W.-K. Ching, Optimal portfolios with regime switching and value-at-risk constraint, Automatica, 46 (2010), 979-989.  doi: 10.1016/j.automatica.2010.02.027. [27] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer New York, 1999. doi: 10.1007/978-1-4612-1466-3. [28] J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM Journal on Control and Optimization, 51 (2013), 2809-2838.  doi: 10.1137/120892477. [29] Z. Yu, Infinite horizon jump-diffusion forward-backward stochastic differential equations and their application to backward linear-quadratic problems, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 1331–1359. doi: 10.1051/cocv/2016055. [30] X. Zhang, R. J. Elliott, T. K. Siu and J. Guo, Markovian regime-switching market completion using additional markov jump assets, IMA Journal of Management Mathematics, 23 (2012), 283–305. doi: 10.1093/imaman/dpr018. [31] X. Zhang and X. Li, Open-Loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems of Markov regime-switching system, arXiv: 1809.01891, 2018. [32] X. Zhang, Z. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM Journal on Control and Optimization, 56 (2018), 2563–2592. doi: 10.1137/17M112395X. [33] 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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