Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises

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.

[1]  O. Costa,et al.  Multiperiod Mean-Variance Optimization with Intertemporal Restrictions , 2007 .

[2]  Berç Rustem,et al.  Multi-period minimax hedging strategies , 1996 .

[3]  Andrew E. B. Lim,et al.  Stochastic optimal LQR control with integral quadratic constraints and indefinite control weights , 1999, IEEE Trans. Autom. Control..

[4]  Oswaldo Luiz V. Costa,et al.  Discrete-time mean variance optimal control of linear systems with Markovian jumps and multiplicative noise , 2009, Int. J. Control.

[5]  Xun Yu Zhou,et al.  Characterizing all optimal controls for an indefinite stochastic linear quadratic control problem , 2002, IEEE Trans. Autom. Control..

[6]  Robert J. Elliott,et al.  State and Mode Estimation for Discrete-Time Jump Markov Systems , 2005, SIAM J. Control. Optim..

[7]  Andrew E. B. Lim,et al.  Discrete time LQG controls with control dependent noise , 1999 .

[8]  Gang George Yin,et al.  Markowitz's Mean-Variance Portfolio Selection with Regime Switching: A Continuous-Time Model , 2003, SIAM J. Control. Optim..

[9]  Markus Leippold,et al.  A Geometric Approach To Multiperiod Mean Variance Optimization of Assets and Liabilities , 2004 .

[10]  Oswaldo Luiz do Valle Costa,et al.  Generalized Coupled Algebraic Riccati Equations for Discrete-time Markov Jump with Multiplicative Noise Systems , 2008, Eur. J. Control.

[11]  X. Chen,et al.  Discrete-time Indefinite LQ Control with State and Control Dependent Noises , 2002, J. Glob. Optim..

[12]  Duan Li,et al.  BETTER THAN DYNAMIC MEAN‐VARIANCE: TIME INCONSISTENCY AND FREE CASH FLOW STREAM , 2012 .

[13]  Xun Yu Zhou,et al.  Indefinite Stochastic Linear Quadratic Control with Markovian Jumps in Infinite Time Horizon , 2003, J. Glob. Optim..

[14]  Süleyman Özekici,et al.  Portfolio optimization in stochastic markets , 2006, Math. Methods Oper. Res..

[15]  Xun Yu Zhou,et al.  Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls , 2000, IEEE Trans. Autom. Control..

[16]  Xun Yu Zhou,et al.  Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon , 2002, Commun. Inf. Syst..

[17]  Jinghao Zhu,et al.  On stochastic Riccati equations for the stochastic LQR problem , 2005, Syst. Control. Lett..

[18]  John B. Moore,et al.  Indefinite Stochastic Linear Quadratic Control and Generalized Differential Riccati Equation , 2002, SIAM J. Control. Optim..

[19]  Oswaldo L. V. Costa,et al.  Mean variance optimal control of Markov jump with multiplicative noise systems , 2007, 2007 European Control Conference (ECC).

[20]  Enmin Feng,et al.  Generalized differential Riccati equation and indefinite stochastic LQ control with cross term , 2004, Appl. Math. Comput..

[21]  A. Stuart,et al.  Portfolio Selection: Efficient Diversification of Investments , 1959 .

[22]  Gang George Yin,et al.  Near-optimal controls of random-switching LQ problems with indefinite control weight costs , 2005, Autom..

[23]  R. P. Marques,et al.  Discrete-Time Markov Jump Linear Systems , 2004, IEEE Transactions on Automatic Control.

[24]  Uwe Mackenroth,et al.  H 2 Optimal Control , 2004 .

[25]  Shu-zhi Wei,et al.  Multi-period optimization portfolio with bankruptcy control in stochastic market , 2007, Appl. Math. Comput..

[26]  Xi Chen,et al.  Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls , 2001, IEEE Trans. Autom. Control..

[27]  Oswaldo Luiz V. Costa,et al.  A generalized multi-period mean-variance portfolio optimization with Markov switching parameters , 2008, Autom..

[28]  Süleyman Özekici,et al.  Portfolio selection in stochastic markets with HARA utility functions , 2010, Eur. J. Oper. Res..

[29]  Xun Yu Zhou,et al.  Stochastic Linear Quadratic Regulators with Indefinite Control Weight Costs. II , 2000, SIAM J. Control. Optim..

[30]  X. Zhou,et al.  Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework , 2000 .

[31]  Shouyang Wang,et al.  Risk control over bankruptcy in dynamic portfolio selection: a generalized mean-variance formulation , 2004, IEEE Transactions on Automatic Control.

[32]  V. Dragan,et al.  The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping , 2004, IEEE Transactions on Automatic Control.

[33]  Oswaldo Luiz V. Costa,et al.  Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems , 2007, Autom..

[34]  B. Rustem,et al.  Robust min}max portfolio strategies for rival forecast and risk scenarios , 2000 .

[35]  Vasile Dragan,et al.  Observability and detectability of a class of discrete-time stochastic linear systems , 2006, IMA J. Math. Control. Inf..

[36]  Marc C. Steinbach,et al.  Markowitz Revisited: Mean-Variance Models in Financial Portfolio Analysis , 2001, SIAM Rev..

[37]  Domenico D'Alessandro,et al.  Discrete-Time Optimal Control with Control-Dependent Noise and Generalized Riccati Difference Equations , 1998, Autom..

[38]  Vasile Dragan,et al.  Mean Square Exponential Stability for some Stochastic Linear Discrete Time Systems , 2006, Eur. J. Control.

[39]  Duan Li,et al.  Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation , 2000 .

[40]  Süleyman Özekici,et al.  Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach , 2007, Eur. J. Oper. Res..

[41]  E. Elton Modern portfolio theory and investment analysis , 1981 .

[42]  V. V. Dombrovskii,et al.  A Linear Quadratic Control for Discrete Systems with Random Parameters and Multiplicative Noise and Its Application to Investment Portfolio Optimization , 2003 .

[43]  Gang George Yin,et al.  Markowitz's mean-variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits , 2004, IEEE Transactions on Automatic Control.

[44]  El-Kébir Boukas,et al.  Stochastic Switching Systems: Analysis and Design , 2005 .

[45]  Berç Rustem,et al.  A robust hedging algorithm , 1997 .