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Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises. (English) Zbl 1260.93173
Summary: In this paper, we consider the stochastic optimal control problem of discrete-time linear systems subject to Markov jumps and multiplicative noises under two criteria. The first one is an unconstrained mean-variance trade-off performance criterion along the time, and the second one is a minimum variance criterion along the time with constraints on the expected output. We present explicit conditions for the existence of an optimal control strategy for the problems, generalizing previous results in the literature. We conclude the paper by presenting a numerical example of a multi-period portfolio selection problem with regime switching in which it is desired to minimize the sum of the variances of the portfolio along the time under the restriction of keeping the expected value of the portfolio greater than some minimum values specified by the investor.
93E20Optimal stochastic control (systems)
93C55Discrete-time control systems
93C05Linear control systems
60J75Jump processes
[1]Rami, M. Ait; Chen, X.; Moore, J. B.; Zhou, X. Y.: Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls, IEEE transactions on automatic control 46, 428-440 (2001) · Zbl 0992.93097 · doi:10.1109/9.911419
[2]Rami, M. Ait; Chen, X.; Zhou, X. Y.: Discrete-time indefinite LQ control with state and control dependent noises, Journal of global optimization 23, 245-265 (2002) · Zbl 1035.49024 · doi:10.1023/A:1016578629272
[3]Rami, M. Ait; Moore, J. B.; Zhou, X. Y.: Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM journal on control and optimization 40, 1296-1311 (2001) · Zbl 1009.93082 · doi:10.1137/S0363012900371083
[4]Rami, M. Ait; Zhou, X. Y.: Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls, IEEE transactions on automatic control 45, 1131-1143 (2000) · Zbl 0981.93080 · doi:10.1109/9.863597
[5]Beghi, A.; D’alessandro, D.: Discrete-time optimal control with control-dependent noise and generalized Riccati difference equations, Automatica 34, 1031-1034 (1998) · Zbl 0944.93032 · doi:10.1016/S0005-1098(98)00044-2
[6]Boukas, E. K.: Stochastic switching systems: analysis and design, (2005)
[7]Cakmak, U.; Ozeckici, S.: Portfolio optimization in stochastic markets, Mathematical methods of operations research 63, 151-168 (2006) · Zbl 1136.91409 · doi:10.1007/s00186-005-0020-x
[8]Canakoglu, E.; Ozeckici, S.: Portfolio selection in stochastic markets with HARA utility functions, European journal of operational research 201, 520-536 (2010) · Zbl 1180.91252 · doi:10.1016/j.ejor.2009.03.017
[9]Celikyurt, U.; Ozeckici, S.: Multperiod portfolio optimization models in stochastic markets using the mean-varaince approach, European journal of operational research 179, 186-202 (2007)
[10]Chen, S.; Li, X.; Zhou, X. Y.: Stochastic linear quadratic regulators with indefinite control weight costs, SIAM journal on control and optimization 36, 1685-1702 (1998) · Zbl 0916.93084 · doi:10.1137/S0363012996310478
[11]Costa, O. L. V.; Araujo, M. V.: A generalized multi-period mean–variance portfolio optimization with Markov switching parameters, Automatica 44, 2487-2497 (2008) · Zbl 1157.91356 · doi:10.1016/j.automatica.2008.02.014
[12]Costa, O. L. V.; De Paulo, W. L.: Indefinite quadratic with linear cost optimal contorl of Markovian jump with multiplicative noise systems, Automatica 43, 587-597 (2007) · Zbl 1115.49021 · doi:10.1016/j.automatica.2006.10.022
[13]Costa, O. L. V.; De Paulo, W. L.: Generalized coupled algebraic Riccati equations for discrete-time Markov jump with multiplicative noise systems, European journal of control 14, 391-408 (2008)
[14]Costa, O. L. V.; Fragoso, M. D.; Marques, R. P.: Discrete-time Markov jump linear systems, (2005)
[15]Costa, O. L. V.; Nabholz, R. B.: Multiperiod mean–variance optimization with intertemporal restrictions, Journal of optimization theory and applications 134, 257-274 (2007) · Zbl 1190.90272 · doi:10.1007/s10957-007-9233-x
[16]Costa, O. L. V.; Okimura, R. T.: Multi-period mean variance optimal control of Markov jump with multiplicative noise systems, Mathematical reports 9, 21-34 (2007) · Zbl 1150.60034
[17]Costa, O. L. V.; Okimura, R. T.: Discrete-time mean variance optimal control of linear systems with Markovian jumps and multiplicative noise, International journal of control 82, 256-267 (2009) · Zbl 1168.93023 · doi:10.1080/00207170802050825
[18]Cui, X. Y., Li, D., Wang, S. Y., & Zhu, S. S. (2010). Better than dynamic mean–variance: time inconsistency and free cash flow stream. Mathematical Finance, doi:10.1111/j.1467-9965.2010.00461.x.
[19]Dombrovskii, V. V.; Lyashenko, E. A.: A linear quadratic control for discrete systems with random parameters and multiplicative noise and its application to investment portfolio optimization, Automatic and remote control 64, 1558-1570 (2003) · Zbl 1061.93097 · doi:10.1023/A:1026057305653
[20]Dragan, V.; Morozan, T.: The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping, IEEE transactions on automatic control 49, 665-675 (2004)
[21]Dragan, V.; Morozan, T.: Mean square exponential stability for some stochastic linear discrete-time systems, European journal of control 12, 373-395 (2006)
[22]Dragan, V.; Morozan, T.: Observability and detectability of a class of discrete-time stochastic linear systems, IMA journal of mathematical control and information 23, 371-394 (2006) · Zbl 1095.93004 · doi:10.1093/imamci/dni064
[23]Elliott, R. J.; Dufour, F.; Malcolm, W. P.: State and mode estimation for discrete-time jump Markov systems, SIAM journal on control and optimization 44, 1081-1104 (2005) · Zbl 1130.93423 · doi:10.1137/S0363012904442628
[24]Elton, E. J.; Gruber, M. J.: Modern portfolio theory and investment analysis, (1995)
[25]Howe, M. A.; Rustem, B.: A robust hedging algorithm, Journal of economic dynamics and control 21, 1065-1092 (1997) · Zbl 0901.90013 · doi:10.1016/S0165-1889(97)00017-1
[26]Howe, M. A.; Rustem, B.; Selby, M. J. P.: Multi-period minimax hedging strategies, European journal of operational research 93, 185-204 (1996) · Zbl 0912.90023 · doi:10.1016/0377-2217(95)00167-0
[27]Leippold, M.; Trojani, F.; Vanini, P.: A geometric approach to multiperiod mean variance optimization of assets and liabilities, Journal of economic dynamics and control 28, 1079-1113 (2004) · Zbl 1179.91234 · doi:10.1016/S0165-1889(03)00067-8
[28]Li, D.; Ng, W. L.: Optimal dynamic portfolio selection: multiperiod mean–variance formulation, Mathematical finance 10, 387-406 (2000) · Zbl 0997.91027 · doi:10.1111/1467-9965.00100
[29]Li, X.; Zhou, X. Y.: Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon, Communications in information and systems 2, 265-282 (2002) · Zbl 1119.93418
[30]Li, X.; Zhou, X. Y.; Rami, M. Ait: Indefinite stochastic linear quadratic control with Markovian jumps in infinite time horizon, Journal of global optimization 27, 149-175 (2003) · Zbl 1031.93155 · doi:10.1023/A:1024887007165
[31]Lim, A.; Zhou, X. Y.: Stochastic optimal control LQR control with integral quadratic constraints and indefinite control weights, IEEE transactions on automatic control 44, 1359-1369 (1999) · Zbl 0970.93038 · doi:10.1109/9.774108
[32]Liu, Y.; Yin, G.; Zhou, X. Y.: Near-optimal controls of random-switching LQ problems with indefinite control weight costs, Automatica 41, 1063-1070 (2005) · Zbl 1091.93019 · doi:10.1016/j.automatica.2005.01.002
[33]Luo, C.; Feng, E.: Generalized differential Riccati equation and indefinite stochastic LQ control with cross term, Applied mathematics and computation 155, 121-135 (2004) · Zbl 1053.93041 · doi:10.1016/S0096-3003(03)00766-5
[34]Markowitz, H.: Portfolio selection: efficient diversification of investments, (1959)
[35]Moore, J. B.; Zhou, X. Y.; Lim, A. E. B.: Discrete-time LQG controls with control dependent noise, Systems control letters 36, 199-206 (1999) · Zbl 0913.93076 · doi:10.1016/S0167-6911(98)00092-9
[36]Rustem, B.; Becker, R. G.; Marty, W.: Robust MIN–MAX portfolio strategies for rival forecast and risk scenarios, Journal of economic dynamics and control 24, 1591-1621 (1995) · Zbl 0967.91026 · doi:10.1016/S0165-1889(99)00088-3
[37]Saberi, A.; Sannuti, P.; Chen, B. M.: H2-optimal control, (1995)
[38]Steinbach, M. C.: Markowitz revisited: mean–variance models in financial portfolio analysis, SIAM review 43, 31-85 (2001) · Zbl 1049.91086 · doi:10.1137/S0036144500376650
[39]Wei, S. Z.; Ye, Z. X.: Multperiod optimization portfolio with bankruptcy control in stochastic market, Applied mathematics and computation 186, 414-425 (2007) · Zbl 1185.91168 · doi:10.1016/j.amc.2006.07.108
[40]Wu, H.; Zhou, X. Y.: Characterizing all optimal controls for an indefinite stochastic linear quadratic control problem, IEEE transactions on automatic control 47, 1119-1122 (2002)
[41]Yin, G.; Zhou, X. Y.: Markowitz’s mean–variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits, IEEE transactions on automatic control 49, 349-360 (2004)
[42]Zhou, X. Y.; Li, D.: Continuous-time mean–variance portfolio selection: a stochastic LQ framework, Applied mathematics optimization 42, 19-33 (2000) · Zbl 0998.91023 · doi:10.1007/s002450010003
[43]Zhou, X. Y.; Yin, G.: Markowitz’s mean–variance portfolio selection with regime switching: a continuous-time model, SIAM journal on control and optimization 42, 1466-1482 (2003) · Zbl 1175.91169 · doi:10.1137/S0363012902405583
[44]Zhu, J.: On stochastic Riccati equations for the stochastic LQR problem, Systems control letters 54, 119-124 (2005)
[45]Zhu, S. S.; Li, D.; Wang, S. Y.: Risk control over bankruptcy in dynamic portfolio selection: a generalized mean variance formulation, IEEE transactions on automatic control 49, 447-457 (2004)