平均场正倒向随机控制系统的最大值原理
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摘要
该文研究具有时间不连续效用函数的平均场随机系统最优控制问题.其中,扩散项系数包含控制变量且控制区域非凸.借助于延拓的Ekeland变分原理及递归方法,建立平均场理论框架下一般形式的随机最大值原理.最后,求解一个线性二次问题以论证结果的可行性.
The present paper concerns with optimal control problems allowing for time inconsistent utility functions for instance of mean-field stochastic systems. Moreover, the control variable enters the diffusion coefficient and the control domain is non-convex. Via extended Ekeland's variational principle as well as the reduction method, a general stochastic maximum principle is established in the framework of mean-field theory. Finally, a linear-quadratic example is worked out to illustrate the application of the results.
The present paper concerns with optimal control problems allowing for time inconsistent utility functions for instance of mean-field stochastic systems. Moreover, the control variable enters the diffusion coefficient and the control domain is non-convex. Via extended Ekeland's variational principle as well as the reduction method, a general stochastic maximum principle is established in the framework of mean-field theory. Finally, a linear-quadratic example is worked out to illustrate the application of the results.
引文
[1] Ansari Q H. Ekeland's variational principle and its extensions with applications//Almezel S, et al. Topics in Fixed Point Theory. Switzerland:Springer International Publishing,2014:65-100
[2] Bensoussan A, Yam S C P, Zhang Z. Well-posedness of mean-field type forward-backward stochastic differential equations. Stoch Process Appl,2015,125:3327-3354
[3] Buckdahn R, Djehiche B, Li J. A general stochastic maximum principle for SDEs of mean-field type. Appl Math Optim, 2011, 64:197-216
[4] Buckdahn R, Djehiche B, Li J, Peng S G. Mean-field backward stochastic differential equations. A limit approach. Ann Probab, 2009, 125:1524-1565
[5] Buckdahn R, Li J, Peng S G. Mean-field backward stochastic differential equations and related partial differential equations. Stoch Process Appl, 2009, 119:3133-3154
[6] Gomes D A,Patrizi S. Weakly coupled mean-field game systems. Nonlinear Analysis, 2016, 144:110-138
[7] Kohlmann M, Zhou X Y. Relationship between backward stochastic differential equations and stochastic controls:a linear-quadratic approach. SIAM J Control Optim, 2000, 38:1392-1407
[8] Lasry J M, Lions P L. Mean field games. Japan J Math, 2007, 2:229-260
[9] Li J. Stochastic maximum principle in the mean-field controls. Automatica, 2012, 48:366-373
[10] Li J. Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integralPDEs. Stoch Process Appl, 2017, http://doi.org/10.1016/j.spa.2017.10.011
[11] Li R J, Liu B. A maximum principle for fully coupled stochastic control systems of mean-field type. J Math Anal Appl, 2014, 415:902-930
[12] Ma H P, Liu B. Maximum principle for partially observed risk-sensitive optimal control problems of meanfield type. Eur J Control, 2016, 32:16-23
[13] Meyer-Brandis T, Aksendal B,Zhou X Y. A mean-field stochastic maximum principle via Malliavin calculus. Stochastics:An International Journal of Probab and Stoch Process,2012, 84:643-666
[14] Ni Y H, Li X, Zhang J F. Mean-field stochastic linear-quadratic optimal control with Markov jump parameters. Systems Control Lett, 2016, 93:69-76
[15] Qi Q Y, Zhang H S. Necessary and sufficient solution to optimal control for linear continuous time meanfield system. IFAC PapersOnLine, 2017, 50:1495-1501
[16] Shen Y, Sui T K. The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem. Nonlinear Analysis, 2013, 86:58-73
[17] Wang G C, Xiao H, Xing G J. An optimal control problem for mean-field forward-backward stochastic differential equation with noisy observation. Automatica, 2017, 86:104-109
[18] Wu Z. A general maximum principle for optimal control of forward-backward stochastic systems. Automatica, 2013, 49:1473-1480
[19] Yong J M,Zhou X Y. Stochastic Controls,Hamiltonian Systems and HJB Equations. New York:SpringerVerlag, 1999
[2] Bensoussan A, Yam S C P, Zhang Z. Well-posedness of mean-field type forward-backward stochastic differential equations. Stoch Process Appl,2015,125:3327-3354
[3] Buckdahn R, Djehiche B, Li J. A general stochastic maximum principle for SDEs of mean-field type. Appl Math Optim, 2011, 64:197-216
[4] Buckdahn R, Djehiche B, Li J, Peng S G. Mean-field backward stochastic differential equations. A limit approach. Ann Probab, 2009, 125:1524-1565
[5] Buckdahn R, Li J, Peng S G. Mean-field backward stochastic differential equations and related partial differential equations. Stoch Process Appl, 2009, 119:3133-3154
[6] Gomes D A,Patrizi S. Weakly coupled mean-field game systems. Nonlinear Analysis, 2016, 144:110-138
[7] Kohlmann M, Zhou X Y. Relationship between backward stochastic differential equations and stochastic controls:a linear-quadratic approach. SIAM J Control Optim, 2000, 38:1392-1407
[8] Lasry J M, Lions P L. Mean field games. Japan J Math, 2007, 2:229-260
[9] Li J. Stochastic maximum principle in the mean-field controls. Automatica, 2012, 48:366-373
[10] Li J. Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integralPDEs. Stoch Process Appl, 2017, http://doi.org/10.1016/j.spa.2017.10.011
[11] Li R J, Liu B. A maximum principle for fully coupled stochastic control systems of mean-field type. J Math Anal Appl, 2014, 415:902-930
[12] Ma H P, Liu B. Maximum principle for partially observed risk-sensitive optimal control problems of meanfield type. Eur J Control, 2016, 32:16-23
[13] Meyer-Brandis T, Aksendal B,Zhou X Y. A mean-field stochastic maximum principle via Malliavin calculus. Stochastics:An International Journal of Probab and Stoch Process,2012, 84:643-666
[14] Ni Y H, Li X, Zhang J F. Mean-field stochastic linear-quadratic optimal control with Markov jump parameters. Systems Control Lett, 2016, 93:69-76
[15] Qi Q Y, Zhang H S. Necessary and sufficient solution to optimal control for linear continuous time meanfield system. IFAC PapersOnLine, 2017, 50:1495-1501
[16] Shen Y, Sui T K. The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem. Nonlinear Analysis, 2013, 86:58-73
[17] Wang G C, Xiao H, Xing G J. An optimal control problem for mean-field forward-backward stochastic differential equation with noisy observation. Automatica, 2017, 86:104-109
[18] Wu Z. A general maximum principle for optimal control of forward-backward stochastic systems. Automatica, 2013, 49:1473-1480
[19] Yong J M,Zhou X Y. Stochastic Controls,Hamiltonian Systems and HJB Equations. New York:SpringerVerlag, 1999