平均场倒向随机微分方程的线性二次最优控制及非零和微分对策
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摘要
本文主要基于正倒向随机微分方程、平均场正倒向随机微分方程和最优控制理论,研究了一类特殊的初始条件耦合的平均场正倒向随机微分方程,然后研究了该类方程的线性二次最优控制以及非零和微分对策问题,最后我们进一步研究了平均场线性正倒向随机系统的近似最优控制问题。
首先我们主要研究了下列初始条件耦合的平均场正倒向随机微分方程在单调性假设条件下解的存在性和唯一性：然后我们给出了上述方程组中正向方程的伴随方程,利用相应的平均场正倒向随机微分方程的解得到了线性二次最优控制问题的最优控制的显式解,也得到了非零和微分对策问题的纳什均衡点的显式解,并且分别证明了它们是唯一的。
其次,我们研究了平均场正倒向线性随机控制系统的近似最优控制问题。结合Zhou [25]我们给出了近似最优控制的定义,证明了近似最优控制的充分性条件和必要性条件。
Based on the theory of forward-backward stochastic differential equations (FBSDEs), mean-field forward-backward stochastic differential equations (mean-field FBSDEs) and the optimal control, we study a new type of initial coupled mean-field FBSDEs. Then, we study mean-field linear quadratic optimal control problems and the nonzero-sum differential games of such equations. In the end we study the near-optimal control problems of mean-field linear forward-backw-ard stochastic systems.
In this paper, firstly, we prove that there exists a unique solution of initial coupled mean-field FBSDEs under the monotonic conditions: Then we give the adjoint equation corresponding to the forward stochastic equa-tion. With the help of the solutions of mean-field FBSDEs, we get the explicit form of the optimal control for the linear quadratic optimal control problems and the open-loop Nash equilibrium point of nonzero-sum differential games.
Secondly, we study the near-optimal control problems of mean-field linear forward-backward stochastic systems. Inspired by Zhou [25] we give the definit-ion of the near-optimality, and establish the sufficient condition and the necess--ary condition of the near-optimality in the form of Pontryagin stochastic maxi--mum principle.
首先我们主要研究了下列初始条件耦合的平均场正倒向随机微分方程在单调性假设条件下解的存在性和唯一性：然后我们给出了上述方程组中正向方程的伴随方程,利用相应的平均场正倒向随机微分方程的解得到了线性二次最优控制问题的最优控制的显式解,也得到了非零和微分对策问题的纳什均衡点的显式解,并且分别证明了它们是唯一的。
其次,我们研究了平均场正倒向线性随机控制系统的近似最优控制问题。结合Zhou [25]我们给出了近似最优控制的定义,证明了近似最优控制的充分性条件和必要性条件。
Based on the theory of forward-backward stochastic differential equations (FBSDEs), mean-field forward-backward stochastic differential equations (mean-field FBSDEs) and the optimal control, we study a new type of initial coupled mean-field FBSDEs. Then, we study mean-field linear quadratic optimal control problems and the nonzero-sum differential games of such equations. In the end we study the near-optimal control problems of mean-field linear forward-backw-ard stochastic systems.
In this paper, firstly, we prove that there exists a unique solution of initial coupled mean-field FBSDEs under the monotonic conditions: Then we give the adjoint equation corresponding to the forward stochastic equa-tion. With the help of the solutions of mean-field FBSDEs, we get the explicit form of the optimal control for the linear quadratic optimal control problems and the open-loop Nash equilibrium point of nonzero-sum differential games.
Secondly, we study the near-optimal control problems of mean-field linear forward-backward stochastic systems. Inspired by Zhou [25] we give the definit-ion of the near-optimality, and establish the sufficient condition and the necess--ary condition of the near-optimality in the form of Pontryagin stochastic maxi--mum principle.
引文
[1]D. Zhang. Forward-backward stochastic differential equations and backw--ard linear quadratic stochastic optimal control problem*. Communicatio--ns in Mathematical Research,2009,25(5):402-410.
[2]Z. Wu. Forward-backward stochastic differential equations, linear quadrat--ic stochastic optimal control problem and nonzero sum differential games*. Journal of Systems Science and Complexity,2005, Apr.
[3]Z. Yu, S. Ji. Linear-quadratic nonzero-sum differential game of backward stochastic equations*. Proceedings of the 27* Chinese Control Conferen-ce. Kunming, Yunnan, China,2008, July 16-18.
[4]J. Li. Stochastic maximum principle in the mean-field controls. Automatica 2012,48:366-373.
[5]S. Peng. A general stochastic maximum principle for optimal control probl--ems. SIAM J. Control and Optimal, Vol.28, No.4, July 1990, pp.966-979.
[6]R. Buckdahn, B. Djehiche, and J. Li. Mean-field backward stochastic diffe--rential equations. A limit approach. Annals of Probability,2009,37(4): 1524-1565.
[7]R. Buckdahn, J. Li, and S. Peng. Mean-field backward stochastic different--ial equations and related patial differential equations. Stochastic Process-es and their Applications,2009,119:3133-3154.
[8]R. Buckdahn, B. Djehiche, and J. Li. A general stochastic maximum princ-iple for SDEs of mean-field type. Appl Math Optim,2011,64:197-216.
[9]Y. Liu, G. Yin and X. Zhou. Near-optimal controls of random-switching LQ problems with indefinite control weight costs. Automatica,2005,41,1063-1070.
[10]E. Pardoux, S. Peng. Adapted solution of a backward stochastic different--ial equation. Systems and Control Letters,1990,14:55-61.
11] S. Peng, Z. Wu. Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim, 1999,37:825-843.
[12]Z. Wu. Maximum principle for optimal control problem of fully couples forward-backward stochastic systems. Systems Sci. Math. Sci,1998,11(3): 249-259.
[13]Y. Han, S. Peng, Z. Wu. Maximum principle for backward doubly stocha-stic control systems with applications. SIAMJ Control. Optim,2010, 4224-4241
[14]Lim, E. B. Andrew and X. Zhou. Stochastic differential equations, linear quadratic stochastic optimal control of backward stochastic differential equations. SIAM Control.Optim,2001,40:450-474.
[15]Y. Hu and S. Peng. Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields,1995,103:273-283.
[16]J. Huang, X. Li, G. Wang. Near-optimal control problems for linear forw-ard-backward stochastic systems. Automatica,2010,46:397-404.
[17]Eddie Hui, J. Huang, X. Li, G. Wang. Near-optimal control for stochastic recursive problems. Systems & Control Letters 60,2011,161-168.
[18]S. Peng. Backward stochastic differential equations and applications to optimal control. Applied Mathematics and Optimization,1993,27:125-144.
[19]J. Yong. A Linear-Quadratic optimal control problem for mean-field stoc-hastic differential equations. NSF Grant DMS-1007514, October 10, 2011.
[20]M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth [J]. Annals of Probability, 2000,28:558-602.
[21]J. P. Lepeltier, J. San Martin. Backward stochastic differential equations with continuous coefficient [J]. Statistics Probability Letters,1997,34: 425-430.
[22]Y. Hu, and S. Peng. Solution of forward-backward stochastic differential equations. Probab. Theory Related Fields,1995,103:273-283.
[23]A.Bensoussan, K.C.J.Sung, S.C.P.Yam, S.P.Yung. Linear-Quadratic mean gield Games. Preprint. Available Online.
[24]S. Peng. Backward stochastic differential equations and applications to optimal control. Applied Mathematics and Optimization,1993,27,125-144.
[25]X. Zhou. Stochastic near-optimal controls:Necessary and Sufficient con--ditions for near-optimality. SIAM Journal on Control and Optimization, 1998,36:929-947.
[26]Hamadene S. Nonzero sum linear-quadratic stochastic differential games and backward-forward equations [J]. Stochastic Anal. Appl.,1999,17:117-130.
[27]R. Buckdahn, J. Li. Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim, 2008,47:444-475.
[28]J. Yong, and X. Zhou. Stochastic controls:Hamiltonian systems and HJB equations. New York:Springer-Verlag,1999.
[29]Ekeland. Non convex minimization problems. Bulletin of Australian Mathematical Society,1979,1,443-474.
[30]J. M. Bismut. An introductory approach to duality in optimal stochast control [J]. SiamRev.,1978,20:62-78.
[31]A. Friedman. Differential Games, Wiley-Interscience, New York,1971.
[32]E. B. Lim and X. Zhou. Linear-quadratic control of backward stochastic differential equations[J]. SIAM J. Control Optim.,2001,40:450-474.
[33]J. Yong. Finding adapted solution of forward-backward stochastic differe--ntial equations method of continuation [J]. Probab. Theory&Rel. Field, 1997,107:537-572.
[34]J. Shi, Z. Wu. The maximum principle for fully coupled forward-backwa--rd stochastic control system. Acta Automat. Sinica,2006,32 (2):161-169.
[35]J. Shi, Z. Wu. Maximum principle for forward-backward stochastic contr--ol system with random jumps and applications to finance. J. Syst. Sci. Complex,2010,23 (2):219-231.
[36]H. Xiao, G. Wang. A necessary condition for optimal control of initial coupled FBSDE with partial information. J.Math.Anal.Appl,2012,386: 412-427.
[37]Z. Yu. Linear quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J. Control,2012,14 (4):1-13.
[38]W. Xu. Stochastic maximum principle for optimal control problem of for--ward and backward system. J. Aust. Math. Soc. Ser. B,1993,37:172-185.
[39]W. M. Wonham. On the separation theorem of stochastic control, SIAMJ. Control Optim.,1968,6:312-326.
[40]A. Bensoussan. Point de Nash dans de cas de fonctionnelles quadratiques et jeux differentiels a N personnes, SIAM J. Control,1974,12(3).
[41]T. Eisele. Nonexistence and nonuniqueness of open-loop equilibria in linear-quadratic differential games, J. Math. Anal. Appl.,1982,37:443-468.
[42]Bernt (?)ksendal. Stochastic Differential Equations.世界图书出版公司,2005.(3)：63-71.
[43]彭实戈,倒向随机微分方程及其应用[J].数学进展,1997,26(2)：97-112.
[44]黄志远,周少普,张子刚,倒向随机微分方程的理论、发展及其应用[J].应用数学,2002,15(2)：9-13.
[45]李登峰,微分对策及其应用.国防工业出版社,2000.
[46]汪嘉冈,现代概率论基础(第二版).复旦大学出版社,2005.
[2]Z. Wu. Forward-backward stochastic differential equations, linear quadrat--ic stochastic optimal control problem and nonzero sum differential games*. Journal of Systems Science and Complexity,2005, Apr.
[3]Z. Yu, S. Ji. Linear-quadratic nonzero-sum differential game of backward stochastic equations*. Proceedings of the 27* Chinese Control Conferen-ce. Kunming, Yunnan, China,2008, July 16-18.
[4]J. Li. Stochastic maximum principle in the mean-field controls. Automatica 2012,48:366-373.
[5]S. Peng. A general stochastic maximum principle for optimal control probl--ems. SIAM J. Control and Optimal, Vol.28, No.4, July 1990, pp.966-979.
[6]R. Buckdahn, B. Djehiche, and J. Li. Mean-field backward stochastic diffe--rential equations. A limit approach. Annals of Probability,2009,37(4): 1524-1565.
[7]R. Buckdahn, J. Li, and S. Peng. Mean-field backward stochastic different--ial equations and related patial differential equations. Stochastic Process-es and their Applications,2009,119:3133-3154.
[8]R. Buckdahn, B. Djehiche, and J. Li. A general stochastic maximum princ-iple for SDEs of mean-field type. Appl Math Optim,2011,64:197-216.
[9]Y. Liu, G. Yin and X. Zhou. Near-optimal controls of random-switching LQ problems with indefinite control weight costs. Automatica,2005,41,1063-1070.
[10]E. Pardoux, S. Peng. Adapted solution of a backward stochastic different--ial equation. Systems and Control Letters,1990,14:55-61.
11] S. Peng, Z. Wu. Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim, 1999,37:825-843.
[12]Z. Wu. Maximum principle for optimal control problem of fully couples forward-backward stochastic systems. Systems Sci. Math. Sci,1998,11(3): 249-259.
[13]Y. Han, S. Peng, Z. Wu. Maximum principle for backward doubly stocha-stic control systems with applications. SIAMJ Control. Optim,2010, 4224-4241
[14]Lim, E. B. Andrew and X. Zhou. Stochastic differential equations, linear quadratic stochastic optimal control of backward stochastic differential equations. SIAM Control.Optim,2001,40:450-474.
[15]Y. Hu and S. Peng. Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields,1995,103:273-283.
[16]J. Huang, X. Li, G. Wang. Near-optimal control problems for linear forw-ard-backward stochastic systems. Automatica,2010,46:397-404.
[17]Eddie Hui, J. Huang, X. Li, G. Wang. Near-optimal control for stochastic recursive problems. Systems & Control Letters 60,2011,161-168.
[18]S. Peng. Backward stochastic differential equations and applications to optimal control. Applied Mathematics and Optimization,1993,27:125-144.
[19]J. Yong. A Linear-Quadratic optimal control problem for mean-field stoc-hastic differential equations. NSF Grant DMS-1007514, October 10, 2011.
[20]M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth [J]. Annals of Probability, 2000,28:558-602.
[21]J. P. Lepeltier, J. San Martin. Backward stochastic differential equations with continuous coefficient [J]. Statistics Probability Letters,1997,34: 425-430.
[22]Y. Hu, and S. Peng. Solution of forward-backward stochastic differential equations. Probab. Theory Related Fields,1995,103:273-283.
[23]A.Bensoussan, K.C.J.Sung, S.C.P.Yam, S.P.Yung. Linear-Quadratic mean gield Games. Preprint. Available Online.
[24]S. Peng. Backward stochastic differential equations and applications to optimal control. Applied Mathematics and Optimization,1993,27,125-144.
[25]X. Zhou. Stochastic near-optimal controls:Necessary and Sufficient con--ditions for near-optimality. SIAM Journal on Control and Optimization, 1998,36:929-947.
[26]Hamadene S. Nonzero sum linear-quadratic stochastic differential games and backward-forward equations [J]. Stochastic Anal. Appl.,1999,17:117-130.
[27]R. Buckdahn, J. Li. Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim, 2008,47:444-475.
[28]J. Yong, and X. Zhou. Stochastic controls:Hamiltonian systems and HJB equations. New York:Springer-Verlag,1999.
[29]Ekeland. Non convex minimization problems. Bulletin of Australian Mathematical Society,1979,1,443-474.
[30]J. M. Bismut. An introductory approach to duality in optimal stochast control [J]. SiamRev.,1978,20:62-78.
[31]A. Friedman. Differential Games, Wiley-Interscience, New York,1971.
[32]E. B. Lim and X. Zhou. Linear-quadratic control of backward stochastic differential equations[J]. SIAM J. Control Optim.,2001,40:450-474.
[33]J. Yong. Finding adapted solution of forward-backward stochastic differe--ntial equations method of continuation [J]. Probab. Theory&Rel. Field, 1997,107:537-572.
[34]J. Shi, Z. Wu. The maximum principle for fully coupled forward-backwa--rd stochastic control system. Acta Automat. Sinica,2006,32 (2):161-169.
[35]J. Shi, Z. Wu. Maximum principle for forward-backward stochastic contr--ol system with random jumps and applications to finance. J. Syst. Sci. Complex,2010,23 (2):219-231.
[36]H. Xiao, G. Wang. A necessary condition for optimal control of initial coupled FBSDE with partial information. J.Math.Anal.Appl,2012,386: 412-427.
[37]Z. Yu. Linear quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J. Control,2012,14 (4):1-13.
[38]W. Xu. Stochastic maximum principle for optimal control problem of for--ward and backward system. J. Aust. Math. Soc. Ser. B,1993,37:172-185.
[39]W. M. Wonham. On the separation theorem of stochastic control, SIAMJ. Control Optim.,1968,6:312-326.
[40]A. Bensoussan. Point de Nash dans de cas de fonctionnelles quadratiques et jeux differentiels a N personnes, SIAM J. Control,1974,12(3).
[41]T. Eisele. Nonexistence and nonuniqueness of open-loop equilibria in linear-quadratic differential games, J. Math. Anal. Appl.,1982,37:443-468.
[42]Bernt (?)ksendal. Stochastic Differential Equations.世界图书出版公司,2005.(3)：63-71.
[43]彭实戈,倒向随机微分方程及其应用[J].数学进展,1997,26(2)：97-112.
[44]黄志远,周少普,张子刚,倒向随机微分方程的理论、发展及其应用[J].应用数学,2002,15(2)：9-13.
[45]李登峰,微分对策及其应用.国防工业出版社,2000.
[46]汪嘉冈,现代概率论基础(第二版).复旦大学出版社,2005.