平均场正倒向随机系统微分对策的最大值原理
|
摘要
自从Buckdahn, Djehiche, Li和Peng [1]首次将平均场引入到倒向随机微分方程(BSDEs)之后,平均场倒向随机微分方程(mean-field backward stochastic differential equations (Mean-field BSDEs))便受到了广大学者的关注。基于此,本文将研究平均场正倒向随机系统的微分对策问题：
其中λv1,v2(t)=(xv1,v2(t),yv1,v2(t),zv1,v2(t))。
在适当的假设下,本文引入相应的代价泛函,研究该平均场正倒向随机系统的微分对策问题：零和以及非零和微分对策的最大值原理。给出了最大值原理的必要条件与充分条件,并且通过给出一非零和微分对策的例子来进一步阐述本文主要定理的应用。
由于一般的随机控制问题可以视为只有一个参赛者的零和微分对策问题,并且在现实生活中,观测者多数情况下只能得到部分信息,所以本文首先研究了部分信息下的平均场正倒向随机微分方程(mean-field forward-backward stochastic differential equations (Mean-field FBSDEs))的最优控制问题,并且利用经典的凸变分技术得到了最优控制的必要条件,其中,随机系统描述如下：
In2009, Buckdahn, Djehiche, Li and Peng [1] firstly introduced the mean-field theory to backward stochastic differential equations (BSDEs), and obtain-ed a new type of backward stochastic differential equations--Mean-field backw-ard stochastic differential equations (Mean-field BSDEs). We will study the di-fferential games of mean-field forward-backward stochastic systems:
where λv1,v2(t)=(xv1,v2(t),yv1,v2(t),zv1,v2(t)).
Under appropriate assumptions, this paper mainly works on the maximum principle for both zero-sum and nonzero-sum games. We give a necessary con--dition and a sufficient condition in the form of maximum principle for the games. In the end, we give an example of a nonzero-sum game of mean-field FBSDE to explain our main results.
Since the general stochastic control problems could be regarded as the zero-sum differential games with only one player, and in reality, the observers can only observe the partial information which is a sub-filtration in probability lang--uage, we will firstly focus on the optimization problems of mean-field FBSDEs with partial information. With the help of the classical convex variational techn--ique, we establish a necessary maximum principle for the optimization proble--ms, where the stochastic system is described as follows:
其中λv1,v2(t)=(xv1,v2(t),yv1,v2(t),zv1,v2(t))。
在适当的假设下,本文引入相应的代价泛函,研究该平均场正倒向随机系统的微分对策问题：零和以及非零和微分对策的最大值原理。给出了最大值原理的必要条件与充分条件,并且通过给出一非零和微分对策的例子来进一步阐述本文主要定理的应用。
由于一般的随机控制问题可以视为只有一个参赛者的零和微分对策问题,并且在现实生活中,观测者多数情况下只能得到部分信息,所以本文首先研究了部分信息下的平均场正倒向随机微分方程(mean-field forward-backward stochastic differential equations (Mean-field FBSDEs))的最优控制问题,并且利用经典的凸变分技术得到了最优控制的必要条件,其中,随机系统描述如下：
In2009, Buckdahn, Djehiche, Li and Peng [1] firstly introduced the mean-field theory to backward stochastic differential equations (BSDEs), and obtain-ed a new type of backward stochastic differential equations--Mean-field backw-ard stochastic differential equations (Mean-field BSDEs). We will study the di-fferential games of mean-field forward-backward stochastic systems:
where λv1,v2(t)=(xv1,v2(t),yv1,v2(t),zv1,v2(t)).
Under appropriate assumptions, this paper mainly works on the maximum principle for both zero-sum and nonzero-sum games. We give a necessary con--dition and a sufficient condition in the form of maximum principle for the games. In the end, we give an example of a nonzero-sum game of mean-field FBSDE to explain our main results.
Since the general stochastic control problems could be regarded as the zero-sum differential games with only one player, and in reality, the observers can only observe the partial information which is a sub-filtration in probability lang--uage, we will firstly focus on the optimization problems of mean-field FBSDEs with partial information. With the help of the classical convex variational techn--ique, we establish a necessary maximum principle for the optimization proble--ms, where the stochastic system is described as follows:
引文
[1]Bismut, J.M. An introductory approach to duality in optimal stochastic control. SIAM. Control and optimization,1978,20:62-78.
[2]Pardoux, E. Peng, S. Adapted solution of a backward stochastic differential equation. Syst. Control Lett,1990,14:55-61.
[3]Wu, Z. The stochastic maximum principle for partially observed forward and backward stochastic control systems. Sci. China, Ser. F,2011.
[4]Wang, G., Wu, Z. Kalman-Bucy filtering equations of forward and backw-ard stochastic systems and applications to recursive optimal control probl--ems. J. Math. Anal. Appl,2008,342:1280-1296.
[5]Wang, G., Wu, Z. The maximum principle for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control, 2009,54(6):1230-1242.
[6]Huang, J., Wang, G., Xiong, J. A maximum principle for partial informati--on backward stochastic control problems with applications. SIAM J. Cont-rol Optim,2009,48(4):2106-2117.
[7]J.M. Lasry, P.L. Lions. Mean field games. Japan. J. Math,2007,2:229-260.
[8]M. Bossy. Some stochastic particle methods for nonlinear parabolic PDEs. ESAIM Proc,2005,15:18-57.
[9]M. Bossy, D. Talay. A stochastic particle method for the McKean-Vlasov and the Burgers equation. Math. Comput,1997,217 (66):157-192.
[10]T. Chan. Dynamics of the McKean-Vlasov equation. Ann. Probab,1994, 22(1):431-441.
[11]P. Kotelenez. A class of quasilinear stochastic partial differential equati--ons of McKean-Vlasov type with mass conservation. Probab. Theory Related Fields,1995,102:159-188.
[12]L. Overbeck. Superprocesses and McKean-Vlasov equations with creati- -on of mass.1995.
[13]P.D. Pra, F.D. Hollander. McKean-Vlasov limit for interacting random processes in random media. J. Statist. Phys,1995,84 (314):735-772.
[14]A.S. Sznitman. Nonlinear reflecting diffusion processes and the propaga-tion of chaos and fluctuations associated. J. Funct. Anal,1984,56:311-336.
[15]A.S. Sznitman. Topics in Propagation of Chaos. Lect. Notes in Math., 1991,1464:165-252.
[16]D. Talay, O. Vaillant. A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations. Ann. Appl. Probab,2003,13(1):140-180.
[17]R. Buckdahn, B. Djehiche, J. Li, S. Peng. Mean-field backward stochastic differential equations. A limit approach. Ann. Probab,2007.
[18]S. Peng. A General Stochastic Maximum Principle for Optimal Control Problems. SIAM J. Control and optimzation,1990,28(4):966-979.
[19]S. Peng, Z. Wu. Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim, 1999,37(3):825-843.
[20]R. Buckdahn, B. Djehiche, J. Li, S. Peng. Mean-field backward stochastic differential equations and related partial differential equations. Stochastic Processes and their Applications,2009,119:3133-3154.
[21]Eddie C.M. Hui, Hua Xiao. Maximum principle for differential games of forward-backward stochastic systems with applications. J.Math. Anal. Appl,2012,386:412-427.
[22]J. Von Neumann, O. Morgenstern. The Theory of Games and Economic Behavior. Princeton University Press, Princeton,1944.
[23]T.T.K. An, B. (?)ksendal. A maximum principle for stochastic differential games with g-expectations and partial information. Preprint Ser. Pure Math,2010, (ISSN 0806-2439) 4.
[24]S. Hamadene. Nonzero-sum linear-quadratic stochastic differential games and backward-forward equations. Stoch. Anal. Appl,1999,17:117-130.
[25]D. Jiang. Equivalent representations of bi-matrix games. Int. J. Innov. Comput. Inform. Control,2009,5 (6):1757-1764.
[26]D. Jiang. Static, completely static, and rational games of complete infor--mation and their different Nash equilibria. Int. J. Innov. Comput. Inform. Control,2008,4 (3):651-660.
[27]M. Jimenez-Lizarraga, L. Fridman, Robust. Nash strategies based on integral sliding mode control for a two players uncertain linear affine-quadratic game. Int. J. Innov. Comput. Inform. Control,2009,5(2):241-252.
[28]M. Jimenez-Lizarraga, M. Basin, Ma. A. Alcorta-Garcia. Equilibrium in linear quadratic stochastic games with unknown parameters. ICIC Express Lett.2009,3(2):107-114.
[29]Y. Konishi, N. Araki, Y. Iwai, H. Ishigaki. Optimal integrated design for mechanical structure and controller using bargaining game theory. ICIC Express Lett,2009,3(1):41-46.
[30]A.E.B. Lim, X. Zhou. Risk-sensitive control with HARA utility. IEEE Trans. Automat. Control,2001,46(4):563-578.
[31]M. J. Osborne. An Introduction to Game Theory. Oxford University Press, 2003.
[32]Juan Li. Stochastic maximum principle in the mean-field controls. Auto--matica,2012,48:366-373.
[33]S. Peng. Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim,1993,27:125-144.
[34]B. (?)ksendal, A. Sulem. Maximum principles for optimal control of forw--ard-backward stochastic differential equations with jumps. SIAM J. Control Optim,2010,48 (5):2945-2976.
[35]J. Shi, Z, Wu. The maximum principle for fully coupled forward-backw--ard stochastic control system. Acta Automat. Sinica,2006,32 (2):161-169.
[36]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.
[37]W. Xu. Stochastic maximum principle for optimal control problem of forward and backward system. J. Aust. Math. Soc. Ser. B,1993,37: 172-185.
[38]J. Yong. A stochastic linear quadratic optimal control problem with gene-ralized expectation. Stoch. Anal. Appl,2008,26:1136-1160.
[39]G. Wang, Z. Yu. A Pontryagin's maximum principle for nonzero- sum dif-ferential games of BSDEs with applications. IEEE Trans. Automat. Control,2010,55 (7):1742-1747.
[40]Z. Yu, S. Ji. Linear-quadratic nonzero- sum differential game of backward stochastic differential equations. Proc.27th Chinese Control Conf., Kun--ming, Yunnan.2008,16-18:562-566.
[41]R. Buckdahn, J. Li. Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim, 2008,47:444-475.
[42]Z. Yu. Linear quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J. Control,2012,14 (4):1-13.
[43]Lim, A.E.B., Zhou, X. Linear-quadratic control of backward stochastic differential equations. SIAM J. Control Optim,2001,40(2):450-474.
[44]Hua Xiao, Guangchen Wang. A necessary condition for optimal control of initial coupled FBSDE with partial information. J.Math. Anal. Appl, 2012,386:412-427.
[45]Bernt (?)ksendal, Stochastic Differential Equations.世界图书也版公司,2006.
[46]李登峰,微分对策及其应用.国防工业出版社,2000。
[2]Pardoux, E. Peng, S. Adapted solution of a backward stochastic differential equation. Syst. Control Lett,1990,14:55-61.
[3]Wu, Z. The stochastic maximum principle for partially observed forward and backward stochastic control systems. Sci. China, Ser. F,2011.
[4]Wang, G., Wu, Z. Kalman-Bucy filtering equations of forward and backw-ard stochastic systems and applications to recursive optimal control probl--ems. J. Math. Anal. Appl,2008,342:1280-1296.
[5]Wang, G., Wu, Z. The maximum principle for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control, 2009,54(6):1230-1242.
[6]Huang, J., Wang, G., Xiong, J. A maximum principle for partial informati--on backward stochastic control problems with applications. SIAM J. Cont-rol Optim,2009,48(4):2106-2117.
[7]J.M. Lasry, P.L. Lions. Mean field games. Japan. J. Math,2007,2:229-260.
[8]M. Bossy. Some stochastic particle methods for nonlinear parabolic PDEs. ESAIM Proc,2005,15:18-57.
[9]M. Bossy, D. Talay. A stochastic particle method for the McKean-Vlasov and the Burgers equation. Math. Comput,1997,217 (66):157-192.
[10]T. Chan. Dynamics of the McKean-Vlasov equation. Ann. Probab,1994, 22(1):431-441.
[11]P. Kotelenez. A class of quasilinear stochastic partial differential equati--ons of McKean-Vlasov type with mass conservation. Probab. Theory Related Fields,1995,102:159-188.
[12]L. Overbeck. Superprocesses and McKean-Vlasov equations with creati- -on of mass.1995.
[13]P.D. Pra, F.D. Hollander. McKean-Vlasov limit for interacting random processes in random media. J. Statist. Phys,1995,84 (314):735-772.
[14]A.S. Sznitman. Nonlinear reflecting diffusion processes and the propaga-tion of chaos and fluctuations associated. J. Funct. Anal,1984,56:311-336.
[15]A.S. Sznitman. Topics in Propagation of Chaos. Lect. Notes in Math., 1991,1464:165-252.
[16]D. Talay, O. Vaillant. A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations. Ann. Appl. Probab,2003,13(1):140-180.
[17]R. Buckdahn, B. Djehiche, J. Li, S. Peng. Mean-field backward stochastic differential equations. A limit approach. Ann. Probab,2007.
[18]S. Peng. A General Stochastic Maximum Principle for Optimal Control Problems. SIAM J. Control and optimzation,1990,28(4):966-979.
[19]S. Peng, Z. Wu. Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim, 1999,37(3):825-843.
[20]R. Buckdahn, B. Djehiche, J. Li, S. Peng. Mean-field backward stochastic differential equations and related partial differential equations. Stochastic Processes and their Applications,2009,119:3133-3154.
[21]Eddie C.M. Hui, Hua Xiao. Maximum principle for differential games of forward-backward stochastic systems with applications. J.Math. Anal. Appl,2012,386:412-427.
[22]J. Von Neumann, O. Morgenstern. The Theory of Games and Economic Behavior. Princeton University Press, Princeton,1944.
[23]T.T.K. An, B. (?)ksendal. A maximum principle for stochastic differential games with g-expectations and partial information. Preprint Ser. Pure Math,2010, (ISSN 0806-2439) 4.
[24]S. Hamadene. Nonzero-sum linear-quadratic stochastic differential games and backward-forward equations. Stoch. Anal. Appl,1999,17:117-130.
[25]D. Jiang. Equivalent representations of bi-matrix games. Int. J. Innov. Comput. Inform. Control,2009,5 (6):1757-1764.
[26]D. Jiang. Static, completely static, and rational games of complete infor--mation and their different Nash equilibria. Int. J. Innov. Comput. Inform. Control,2008,4 (3):651-660.
[27]M. Jimenez-Lizarraga, L. Fridman, Robust. Nash strategies based on integral sliding mode control for a two players uncertain linear affine-quadratic game. Int. J. Innov. Comput. Inform. Control,2009,5(2):241-252.
[28]M. Jimenez-Lizarraga, M. Basin, Ma. A. Alcorta-Garcia. Equilibrium in linear quadratic stochastic games with unknown parameters. ICIC Express Lett.2009,3(2):107-114.
[29]Y. Konishi, N. Araki, Y. Iwai, H. Ishigaki. Optimal integrated design for mechanical structure and controller using bargaining game theory. ICIC Express Lett,2009,3(1):41-46.
[30]A.E.B. Lim, X. Zhou. Risk-sensitive control with HARA utility. IEEE Trans. Automat. Control,2001,46(4):563-578.
[31]M. J. Osborne. An Introduction to Game Theory. Oxford University Press, 2003.
[32]Juan Li. Stochastic maximum principle in the mean-field controls. Auto--matica,2012,48:366-373.
[33]S. Peng. Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim,1993,27:125-144.
[34]B. (?)ksendal, A. Sulem. Maximum principles for optimal control of forw--ard-backward stochastic differential equations with jumps. SIAM J. Control Optim,2010,48 (5):2945-2976.
[35]J. Shi, Z, Wu. The maximum principle for fully coupled forward-backw--ard stochastic control system. Acta Automat. Sinica,2006,32 (2):161-169.
[36]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.
[37]W. Xu. Stochastic maximum principle for optimal control problem of forward and backward system. J. Aust. Math. Soc. Ser. B,1993,37: 172-185.
[38]J. Yong. A stochastic linear quadratic optimal control problem with gene-ralized expectation. Stoch. Anal. Appl,2008,26:1136-1160.
[39]G. Wang, Z. Yu. A Pontryagin's maximum principle for nonzero- sum dif-ferential games of BSDEs with applications. IEEE Trans. Automat. Control,2010,55 (7):1742-1747.
[40]Z. Yu, S. Ji. Linear-quadratic nonzero- sum differential game of backward stochastic differential equations. Proc.27th Chinese Control Conf., Kun--ming, Yunnan.2008,16-18:562-566.
[41]R. Buckdahn, J. Li. Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim, 2008,47:444-475.
[42]Z. Yu. Linear quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J. Control,2012,14 (4):1-13.
[43]Lim, A.E.B., Zhou, X. Linear-quadratic control of backward stochastic differential equations. SIAM J. Control Optim,2001,40(2):450-474.
[44]Hua Xiao, Guangchen Wang. A necessary condition for optimal control of initial coupled FBSDE with partial information. J.Math. Anal. Appl, 2012,386:412-427.
[45]Bernt (?)ksendal, Stochastic Differential Equations.世界图书也版公司,2006.
[46]李登峰,微分对策及其应用.国防工业出版社,2000。