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Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems. (English) Zbl 1115.49021

Summary: In this paper we consider the stochastic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The performance criterion is assumed to be formed by a linear combination of a quadratic part and a linear part in the state and control variables. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a necessary and sufficient condition under which the problem is well posed and a state feedback solution can be derived from a set of coupled generalized Riccati difference equations interconnected with a set of coupled linear recursive equations. For the case in which the quadratic-term matrices are non-negative, this necessary and sufficient condition can be written in a more explicit way. The results are applied to a problem of portfolio optimization.

MSC:

49K45 Optimality conditions for problems involving randomness
49K40 Sensitivity, stability, well-posedness
93E20 Optimal stochastic control
91B28 Finance etc. (MSC2000)
60J05 Discrete-time Markov processes on general state spaces
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[1] Ait Rami, M.; 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
[2] Ait Rami, M.; 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
[3] Ait Rami, M.; Moore, J. B.; Zhou, X. Y., Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM Journal on Control Optimization, 40, 1296-1311 (2001) · Zbl 1009.93082
[4] Ait Rami, M.; 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
[5] Basin, M.; Perez, J.; Skliar, M., Optimal filtering for polynomial system states with polynomial multiplicative noise, International Journal of Robust and Nonlinear Control, 16, 303-314 (2006) · Zbl 1105.93056
[6] Basin, M.; Perez, J.; Skliar, M., Optimal state filtering and parameter identification for linear systems, (Proceedings of the 2006 American control conference (2006), Minneapolis: Minneapolis Minnesota), 987-990
[7] 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
[8] Blair, W. P.; Sworder, D. D., Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria, International Journal of Control, 21, 833-841 (1975) · Zbl 0303.93084
[9] Chen, S.; Li, X.; Zhou, X. Y., Stochastic linear quadratic regulators with indefinite control weight costs, SIAM Journal on Control Optimization, 36, 1685-1702 (1998) · Zbl 0916.93084
[10] Costa, O. L.V.; Fragoso, M. D.; Marques, R. P., Discrete-time Markov jump linear systems (2005), Springer: Springer Berlin · Zbl 1081.93001
[11] Costa, O. L.V.; Kubrusly, C. S., State-feedback \(H_\infty \)-control for discrete-time infinite-dimensional stochastic bilinear systems, Journal of Mathematical Systems, Estimation Control, 6, 1-32 (1996) · Zbl 0844.93036
[12] Dombrovskii, V. V.; Dombrovskii, D. V.; Lyashenko, E. A., Predictive control of random-parameter systems with multiplicative noise. Application to investment portfolio optimization, Automation and Remote Control, 66, 583-595 (2005) · Zbl 1114.93099
[13] 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
[14] Dragan, V.; Morozan, T., Stability and robust stabilization to linear stochastic systems described by differential equations with Markovian jumping and multiplicative white noise, Stochastic Analysis and Applications, 20, 33-92 (2002) · Zbl 1136.60335
[15] 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) · Zbl 1365.93541
[16] Gershon, E.; Shaked, U., Static \(H_2\) and \(H_\infty\) output-feedback of discrete-time LTI systems with state multiplicative noise, Systems & Control Letters, 55, 232-239 (2006) · Zbl 1129.93372
[17] Li, D.; Ng, W.-L., Optimal dynamic portfolio selection: Multiperiod mean-variance formulation, Mathematical Finance, 10, 387-406 (2000) · Zbl 0997.91027
[18] 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
[19] Li, X.; Zhou, X. Y.; AitRami, M., Indefinite stochastic linear quadratic control with Markovian jumps in infinite time horizon, Journal of Global Optimization, 27, 149-175 (2003) · Zbl 1031.93155
[20] 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
[21] 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
[22] 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
[23] 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
[24] Ran, A. C.M.; Trentelman, H. L., Linear quadratic problems with indefinite cost for discrete time systems, SIAM Journal on Matrix Analysis and Applications, 14, 776-797 (1993) · Zbl 0786.93043
[25] Roll, R., A mean/variance analysis of tracking error, Journal of Portfolio Management, 18, 13-22 (1992)
[26] Rudolf, M.; Wolter, H. J.; Zimmermann, H., A linear model for tracking error minimization, Journal of Banking and Finance, 23, 85-103 (1999)
[27] Saberi, A.; Sannuti, P.; Chen, B. M., \(H_2\)-Optimal control (1995), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0876.93001
[28] 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) · Zbl 1364.49044
[29] 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) · Zbl 1366.91148
[30] Zhou, X. Y.; Li, D., Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42, 19-33 (2000) · Zbl 0998.91023
[31] 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
[32] Zhu, J., On stochastic Riccati equations for the stochastic LQR problem, Systems & Control Letters, 54, 119-124 (2005) · Zbl 1129.93549
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