带跳跃平均场倒向随机微分方程的线性二次最优控制

Abstract

Abstract:

This paper is concerned with a linear quadratic optimal control problem for meanfield backward stochastic differential equations driven by a Poisson random martingale measure and a Brownian motion. Firstly, by the classic convex variation principle, the existence and uniqueness of the optimal control is obtained. Secondly, the optimal control is characterized by the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps by the duality method. Thirdly, in terms of a decoupling technique, the stochastic Hamilton system is decoupled by introducing two Riccati equations and a MF-BSDE with jumps. Then an explicit representation for the optimal control is obtained.

Key words: Mean-field, Optimal control, Backward stochastic Differential equation, Adjoint process

CLC Number:

O232

Maoning Tang,Qingxin Meng. Linear-Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations with Jumps[J].Acta mathematica scientia,Series A, 2019, 39(3): 620-637.

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References 29

1	Andersson D , Djehiche B . A maximum principle for SDEs of mean-field type. Applied Mathematics and Optimization, 2011, 63: 341- 356 doi: 10.1007/s00245-010-9123-8
2	Baghery F , Baghery F , Øksendal B . A maximum principle for stochastic control with partial information. Stochastic Analysis and Applications, 2007, 25 (3): 705- 717 doi: 10.1080/07362990701283128
3	Buckdahn R , Djehiche B , Li J . A general stochastic maximum principle for SDEs of mean-field type. Applied Mathematics and Optimization, 2011, 64: 197- 216 doi: 10.1007/s00245-011-9136-y
4	Buckdahn R , Djehiche B , Li J , Peng S . Mean-field backward stochastic differential equations:a limit approach. The Annals of Probability, 2009, 37 (4): 1524- 1565 doi: 10.1214/08-AOP442
5	Buckdahn R , Li J , Peng S . Mean-field backward stochastic differential equations and related partial differential equations. Stochastic Processes and Their Applications, 2009, 119 (10): 3133- 3154 doi: 10.1016/j.spa.2009.05.002
6	Du H , Huang J , Qin Y . A stochastic maximum principle for delayed mean-field stochastic differential equations and its applications. IEEE Transactions on Automatic Control, 2013, 38: 3212- 3217
7	Ekeland I , Témam R . Convex Analysis and Variational Problems. Amsterdam: North-Holland, 1976
8	Elliott R , Li X , Ni Y H . Discrete time mean-field stochastic linear-quadratic optimal control problems. Automatica, 2013, 49 (11): 3222- 3233 doi: 10.1016/j.automatica.2013.08.017
9	Hafayed M . A mean-field maximum principle for optimal control of forward-backward stochastic differential equations with Poisson jump processes. International Journal of Dynamics and Control, 2013, 1 (4): 300- 315 doi: 10.1007/s40435-013-0027-8
10	Huang J , Li X , Yong J . A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control and Related Fields, 2015, 5 (1): 97- 139 doi: 10.3934/mcrf
11	Li X, Sun J, Xiong J. Linear quadratic optimal control problems for mean-field backward stochastic differential equations. Applied Mathematics and Optimization.[2017-12-07]. http://doi.org/10.1007/s00245-017-9464-7
12	Li J . Stochastic maximum principle in the mean-field controls. Automatica, 2012, 48 (2): 366- 373 doi: 10.1016/j.automatica.2011.11.006
13	Ma H , Liu B . Linear quadratic optimal control problem for partially observed forward backward stochastic differential equations of mean-field type. Asian Journal of Control, 2017, 19 (1): 1- 12
14	Ma L , Zhang W . Output feedback H^∞ control for discrete time mean-field stochastic systems. Asian Journal of Control, 2015, 17 (6): 2241- 2251 doi: 10.1002/asjc.1128
15	Ma L , Zhang T , Zhang W . H^∞ control for continuous time mean-field stochastic systems. Asian Journal of Control, 2016, 18 (5): 1630- 1640 doi: 10.1002/asjc.1290
16	Meng Q , Shen Y . Optimal control of mean-field jump-diffusion systems with delay:A stochastic maximum principle approach. Journal of Computational and Applied Mathematics, 2015, 279: 13- 30 doi: 10.1016/j.cam.2014.10.011
17	Meyer-Brandis T , Øksendal B , Zhou X Y . A mean-field stochastic maximum principle via Malliavin calculus. Stochastics, 2012, 84: 643- 666 doi: 10.1080/17442508.2011.651619
18	Ni Y H , Li X , Zhang J F . Finite-horizon indefinite mean-field stochastic linear-quadratic optimal control. IFAC-PapersOnLine, 2015, 48 (28): 211- 216 doi: 10.1016/j.ifacol.2015.12.127
19	Ni Y H , Zhang J F , Li X . Indefinite mean-field stochastic linear-quadratic optimal control. IEEE Transactions on Automatic Control, 2015, 60 (7): 1786- 1800 doi: 10.1109/TAC.2014.2385253
20	Zhang H, Qi Q. A Complete solution to optimal control and stabilization for mean-field systems: Part I, Discrete-time case. 2016, arXiv: 1608.06363
21	Qi Q, Zhang H. A Complete solution to optimal control and stabilization for mean-field systems: Part Ⅱ, Continuous-time case. 2016, arXiv: 1608.06475
22	Shen Y , Meng Q , Shi P . Maximum principle for mean-field jump diffusion stochastic delay differential equations and its application to finance. Automatica, 2014, 50 (6): 1565- 1579 doi: 10.1016/j.automatica.2014.03.021
23	Shen Y , Siu T K . The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem. Nonlinear Analysis:Theory, Methods and Applications, 2013, 86: 58- 73 doi: 10.1016/j.na.2013.02.029
24	Tang M, Meng Q. Linear-quadratic optimal control problems for mean-field stochastic differential equations with jumps. 2016, arXiv: 1610.03193
25	Wang G, Wu Z, Zhang C. Maximum principles for partially observed mean-field stochastic systems with application to financial engineering. 2014, DOI: 10.1109/ChiCC.2014.6895853
26	Wang G, Wu Z, Zhang C. A partially observed optimal control problem for mean-field type forwardbackward stochastic system. 2016, DOI: 10.1109/ChiCC.2016.7553351
27	Wang G, Xiao H, Xing G. A class of optimal control problems for mean-field forward-backward stochastic systems with partial information. 2015, arXiv: 1509.03729
28	Wang G , Zhang C , Zhang W . Stochastic maximum principle for mean-field type optimal control under partial information. IEEE Transactions on Automatic Control, 2014, 59 (2): 522- 528 doi: 10.1109/TAC.2013.2273265
29	Yong J . Linear-quadratic optimal control problems for mean-field stochastic differential equations. SIAM Journal on Control and Optimization, 2013, 51 (4): 2809- 2838 doi: 10.1137/120892477

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[15]	Guo Zijun; Wu Rangquan. The Comparison Theorem for Solutions of Forward-backward Stochastic Differential Equations [J]. Acta mathematica scientia,Series A, 2007, 27(2): 368-373.

Linear-Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations with Jumps

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