平均场正倒向随机控制系统的最大值原理

Abstract

Abstract:

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.

Key words: Mean-field SDE, Maximum principle, Adjoint equation, Extended Ekeland's variational principle

CLC Number:

O110.64

Ruijing Li. A General Maximum Principle for Forward-Backward Stochastic Control Systems of Mean-Field Type[J].Acta mathematica scientia,Series A, 2019, 39(1): 143-155.

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References 19

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 doi: 10.1016/j.spa.2015.04.006
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 doi: 10.1007/s00245-011-9136-y
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 doi: 10.1016/j.spa.2009.05.002
6	Gomes D A , Patrizi S . Weakly coupled mean-field game systems. Nonlinear Analysis, 2016, 144: 110- 138 doi: 10.1016/j.na.2016.05.017
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 doi: 10.1137/S036301299834973X
8	Lasry J M , Lions P L . Mean field games. Japan J Math, 2007, 2: 229- 260 doi: 10.1007/s11537-007-0657-8
9	Li J . Stochastic maximum principle in the mean-field controls. Automatica, 2012, 48: 366- 373 doi: 10.1016/j.automatica.2011.11.006
10	Li J . Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integralPDEs. Stoch Process Appl, 2017 doi: 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 doi: 10.1016/j.jmaa.2014.02.008
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 doi: 10.1016/j.ejcon.2016.05.002
13	Meyer-Brandis T , Ãksendal 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 doi: 10.1080/17442508.2011.651619
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 doi: 10.1016/j.sysconle.2016.04.002
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 doi: 10.1016/j.ifacol.2017.08.298
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 doi: 10.1016/j.na.2013.02.029
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 doi: 10.1016/j.automatica.2017.07.018
18	Wu Z . A general maximum principle for optimal control of forward-backward stochastic systems. Automatica, 2013, 49: 1473- 1480 doi: 10.1016/j.automatica.2013.02.005
19	Yong J M , Zhou X Y . Stochastic Controls, Hamiltonian Systems and HJB Equations. New York: SpringerVerlag, 1999

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A General Maximum Principle for Forward-Backward Stochastic Control Systems of Mean-Field Type

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