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Characterization of optimal feedback for stochastic linear quadratic control problems
School of Mathematics, Sichuan University, Chengdu 610064, Sichuan Province, China 
References:
[1] 
Ait Rami, M, Moore, JB, Zhou, X:Indefinite stochastic linear quadratic control and generalized differential Riccati equation. SIAM J. Control Optim 40, 12961311 (2001) 
[2] 
Athans, M:The role and use of the stochastic linearquadraticGaussian problem in control system design.IEEE Trans. Automat. Control 16, 529552 (1971) 
[3] 
BenIsrael, A, Greville, TNE:Generalized Inverses:Theory and Applications. Pure and Applied Mathematics. WileyInterscience[John Wiley & Sons], New YorkLondonSydney (1974) 
[4] 
Bensoussan, A:Lectures on stochastic control. In:Nonlinear Filtering and Stochastic Control. Lecture Notes in Math, vol. 972, pp. 162. SpringerVerlag, Berlin (1981) 
[5] 
Bismut, JM:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim 14, 419444 (1976) 
[6] 
Bismut, JM:Contrôle des systèmes linéaires quadratiques:applications de l'intégrale stochastique. In:Séminaire de Probabilités XII, Université de Strasbourg 1976/77, Lecture Notes in Math, vol. 649, pp. 180264. SpringerVerlag, Berlin (1978) 
[7] 
Briand, PH, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:L^{p} solutions of backward stochastic differential equations. Stochastic Process. Appl 108, 109129 (2003) 
[8] 
Chen, S, Li, X, Zhou, X:Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J. Control Optim 36, 16851702 (1998) 
[9] 
Davis, MHA:Linear Estimation and Stochastic Control. Chapman and Hall Mathematics Series. Chapman and Hall, London; Halsted Press[John Wiley & Sons], New York (1977) 
[10] 
Delbaen, F, Tang, S:Harmonic analysis of stochastic equations and backward stochastic differential equations. Probab. Theory Relat. Fields 146, 291336 (2010) 
[11] 
Frei, C, dos Reis, G:A financial market with interacting investors:does an equilibrium exist? Math. Finan.Econ 4, 161182 (2011) 
[12] 
Kalman, RE:Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5, 102119 (1960) 
[13] 
Lü, Q, Zhang, X:Wellposedness of backward stochastic differential equations with general filtration. J. Diff. Equations 254, 32003227 (2013) 
[14] 
Lü, Q, Zhang, X:General PontryaginType Stochastic Maximum Principle and Backward StochasticEvolution Equations in Infinite Dimensions. Springer Briefs in Mathematics. Springer, Cham (2014) 
[15] 
Lü, Q, Zhang, X:Transposition method for backward stochastic evolution equations revisited, and its application. Math. Control Relat. Fields 5, 529555 (2015) 
[16] 
Lü, Q, Zhang, X:Optimal feedback for stochastic linear quadratic control and backward stochastic Riccati equations in infinite dimensions. (2017). Preprint Pardoux, E, Peng, S:Adapted solution of backward stochastic equation. Systems Control Lett 14, 5561(1990) 
[17] 
Peng, S:Stochastic HamiltonJacobiBellman equations. SIAM J. Control Optim 30, 284304 (1992) 
[18] 
Pham, H:Linear quadratic optimal control of conditional McKeanVlasov equation with random coefficients and applications. (2017). arXiv:1604.06609v1 
[19] 
Protter, PE:Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, vol. 21. SpringerVerlag, Berlin (2005) 
[20] 
Reid, WT:A matrix differential equation of Riccati type. Amer. J. Math 68, 237246 (1946) 
[21] 
Sun, J, Yong, J:Linear quadratic stochastic differential games:openloop and closedloop saddle points.SIAM J. Control Optim 52, 40824121 (2014) 
[22] 
Tang, S:General linear quadratic optimal stochastic control problems with random coefficients:linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim 42, 5375 (2003) 
[23] 
Tang, S:Dynamic programming for general linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim 53, 10821106 (2015) 
[24] 
Wonham, WM:On a matrix Riccati equation of stochastic control. SIAM J. Control 6, 681697 (1968) 
[25] 
Wonham, WM:Linear Multivariable Control, a Geometric Approach. Applications of Mathematics, vol. 10. SpringerVerlag, New York (1985) 
[26] 
Yong, J, Lou, H:A Concise Course on Optimal Control Theory. Higher Education Press, Beijing (2006).(In Chinese) 
[27] 
Yong, J, Zhou, XY:Stochastic Controls:Hamiltonian Systems and HJB Equations. SpringerVerlag, New York, Berlin (2000) 
show all references
References:
[1] 
Ait Rami, M, Moore, JB, Zhou, X:Indefinite stochastic linear quadratic control and generalized differential Riccati equation. SIAM J. Control Optim 40, 12961311 (2001) 
[2] 
Athans, M:The role and use of the stochastic linearquadraticGaussian problem in control system design.IEEE Trans. Automat. Control 16, 529552 (1971) 
[3] 
BenIsrael, A, Greville, TNE:Generalized Inverses:Theory and Applications. Pure and Applied Mathematics. WileyInterscience[John Wiley & Sons], New YorkLondonSydney (1974) 
[4] 
Bensoussan, A:Lectures on stochastic control. In:Nonlinear Filtering and Stochastic Control. Lecture Notes in Math, vol. 972, pp. 162. SpringerVerlag, Berlin (1981) 
[5] 
Bismut, JM:Linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim 14, 419444 (1976) 
[6] 
Bismut, JM:Contrôle des systèmes linéaires quadratiques:applications de l'intégrale stochastique. In:Séminaire de Probabilités XII, Université de Strasbourg 1976/77, Lecture Notes in Math, vol. 649, pp. 180264. SpringerVerlag, Berlin (1978) 
[7] 
Briand, PH, Delyon, B, Hu, Y, Pardoux, E, Stoica, L:L^{p} solutions of backward stochastic differential equations. Stochastic Process. Appl 108, 109129 (2003) 
[8] 
Chen, S, Li, X, Zhou, X:Stochastic linear quadratic regulators with indefinite control weight costs. SIAM J. Control Optim 36, 16851702 (1998) 
[9] 
Davis, MHA:Linear Estimation and Stochastic Control. Chapman and Hall Mathematics Series. Chapman and Hall, London; Halsted Press[John Wiley & Sons], New York (1977) 
[10] 
Delbaen, F, Tang, S:Harmonic analysis of stochastic equations and backward stochastic differential equations. Probab. Theory Relat. Fields 146, 291336 (2010) 
[11] 
Frei, C, dos Reis, G:A financial market with interacting investors:does an equilibrium exist? Math. Finan.Econ 4, 161182 (2011) 
[12] 
Kalman, RE:Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 5, 102119 (1960) 
[13] 
Lü, Q, Zhang, X:Wellposedness of backward stochastic differential equations with general filtration. J. Diff. Equations 254, 32003227 (2013) 
[14] 
Lü, Q, Zhang, X:General PontryaginType Stochastic Maximum Principle and Backward StochasticEvolution Equations in Infinite Dimensions. Springer Briefs in Mathematics. Springer, Cham (2014) 
[15] 
Lü, Q, Zhang, X:Transposition method for backward stochastic evolution equations revisited, and its application. Math. Control Relat. Fields 5, 529555 (2015) 
[16] 
Lü, Q, Zhang, X:Optimal feedback for stochastic linear quadratic control and backward stochastic Riccati equations in infinite dimensions. (2017). Preprint Pardoux, E, Peng, S:Adapted solution of backward stochastic equation. Systems Control Lett 14, 5561(1990) 
[17] 
Peng, S:Stochastic HamiltonJacobiBellman equations. SIAM J. Control Optim 30, 284304 (1992) 
[18] 
Pham, H:Linear quadratic optimal control of conditional McKeanVlasov equation with random coefficients and applications. (2017). arXiv:1604.06609v1 
[19] 
Protter, PE:Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, vol. 21. SpringerVerlag, Berlin (2005) 
[20] 
Reid, WT:A matrix differential equation of Riccati type. Amer. J. Math 68, 237246 (1946) 
[21] 
Sun, J, Yong, J:Linear quadratic stochastic differential games:openloop and closedloop saddle points.SIAM J. Control Optim 52, 40824121 (2014) 
[22] 
Tang, S:General linear quadratic optimal stochastic control problems with random coefficients:linear stochastic Hamilton systems and backward stochastic Riccati equations. SIAM J. Control Optim 42, 5375 (2003) 
[23] 
Tang, S:Dynamic programming for general linear quadratic optimal stochastic control with random coefficients. SIAM J. Control Optim 53, 10821106 (2015) 
[24] 
Wonham, WM:On a matrix Riccati equation of stochastic control. SIAM J. Control 6, 681697 (1968) 
[25] 
Wonham, WM:Linear Multivariable Control, a Geometric Approach. Applications of Mathematics, vol. 10. SpringerVerlag, New York (1985) 
[26] 
Yong, J, Lou, H:A Concise Course on Optimal Control Theory. Higher Education Press, Beijing (2006).(In Chinese) 
[27] 
Yong, J, Zhou, XY:Stochastic Controls:Hamiltonian Systems and HJB Equations. SpringerVerlag, New York, Berlin (2000) 
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