您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2021, Vol. 56 ›› Issue (10): 1-10.doi: 10.6040/j.issn.1671-9352.9.2021.012

• •    

正倒向随机最优控制问题的最大值原理:完全信息和部分信息

吴臻1*,王光臣2,李敏1   

  1. 1.山东大学数学学院, 山东 济南 250100;2.山东大学控制科学与工程学院, 山东 济南 250061
  • 发布日期:2021-09-28
  • 作者简介:吴臻(1971— ), 男, 博士, 教授, 博士生导师, 研究方向为随机控制、 正倒向随机微分方程、 数理金融. E-mail:wuzhen@sdu.edu.cn*通信作者
  • 基金资助:
    国家自然科学基金重点资助项目(11831010);中港合作项目(61961160732);国家自然科学基金杰出青年科学基金资助项目(61925306);山东省自然基金重大项目(ZR2019ZD42,ZR2020ZD24);山东省泰山学者攀登计划(TSPD20210302)

Maximum principle for optimal control of forward-backward stochastic system: full information and partial information

WU Zhen1*, WANG Guang-chen2, LI Min1   

  1. 1. School of Mathematics, Shandong University, Jinan 250100, Shandong, China;
    2. School of Control Science and Engineering, Shandong University, Jinan 250061, Shandong, China
  • Published:2021-09-28

PDF (PC)

606

摘要： 本文是一篇关于正倒向随机最优控制问题研究进展的综述论文。近30年来,正倒向随机控制系统的各种理论与应用研究得到了迅猛发展,取得了大量原创性科研成果,吸引了大批国际同行跟进研究。限于论文篇幅和作者意图,本文仅仅聚焦于正倒向随机最优控制问题的最大值原理这一主题,概述其最新研究进展及其在求解线性二次最优控制问题中的简单应用。

关键词: 正倒向随机系统, 一般最大值原理, 最优滤波, 倒向分离方法, 线性二次控制

Abstract: This paper reviews some progresses on forward-backward stochastic control system. In the past 30 years, various theories and applications of forward-backward stochastic control system have been developed rapidly and a large number of original scientific research results have been obtained, which have attracted many international peers to follow-up research. Limited to the length of this paper and the authors emphasis, this paper only focuses on maximum principle for optimal control of forward-backward stochastic system, and summarizes its latest research progress and application in solving linear quadratic optimal control problem.

Key words: forward-backward stochastic system, general maximum principle, optimal filtering, backward separation approach, linear-quadratic control

中图分类号: 

  • O211
[1] BISMUT J M. Linear quadratic optimal stochastic control with random coefficients[J]. SIAM Journal on Control and Optimization, 1976, 14(3):419-444.
[2] PARDOUX E, PENG S G. Adapted solution of a backward stochastic differential equation[J]. Systems & Control Letters, 1990, 14(1):55-61.
[3] ANTONELLI F. Backward-forward stochastic differential equations[J]. The Annals of Applied Probability, 1993, 3(3):777-793.
[4] PARDOUX E, TANG S J. Forward-backward stochastic differential equations and quasilinear parabolic PDEs[J]. Probability Theory and Related Fields, 1999, 114(2):123-150.
[5] MA J, PROTTER P, YONG J M. Solving forward-backward stochastic differential equations explicitly: a four step scheme[J]. Probability Theory and Related Fields, 1994, 98(3):339-359.
[6] HU Y, PENG S G. Solution of forward-backward stochastic differential equations[J]. Probability Theory and Related Fields, 1995, 103(2):273-283.
[7] PENG S G, WU Z. Fully coupled forward-backward stochastic differential equations and applications to optimal control[J]. SIAM Journal on Control and Optimization, 1999, 37(3):825-843.
[8] MA J, YONG J M. Forward-backward stochastic differential equations and their applications[M] //Lecture Notes in Math 1702. New York: Springer-Verlag, 1999.
[9] YONG J M, ZHOU X Y. Stochastic controls: Hamiltonian systems and HJB euqations[M]. New York: Springer-Verlag, 1999.
[10] MA J, WU Z, ZHANG D T, et al. On well-posedness of forward-backward SDEs: a unified approach[J]. The Annals of Applied Probability, 2015, 25(4):2168-2214.
[11] WANG G C, WU Z. The maximum principles for stochastic recursive optimal control problems under partial information[J]. IEEE Transactions on Automatic Control, 2009, 54(6):1230-1242.
[12] PENG S G. A general stochastic maximum principle for optimal control problems[J]. SIAM Journal on Control and Optimization, 1990, 28(4):966-979.
[13] PENG S G. Backward stochastic differential equations and applications to optimal control[J]. Applied Mathematics and Optimization, 1993, 27(2):125-144.
[14] XU W S. Stochastic maximum principle for optimal control problem of forward and backward system[J]. The ANZIAM Journal, 1995, 37(2):172-185.
[15] WU Z. Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems[J]. Journal of Mathematics and System Science. 1998, 11(3):249-259.
[16] PENG S G. Open problems on backward stochastic differential equations[M] // Control of Distributed Parameter and Stochastic Systems. Boston: Springer, 1999: 265-273.
[17] WU Z. A general maximum principle for optimal control of forward-backward stochastic systems[J]. Automatica, 2013, 49(5):1473-1480.
[18] HU M S. Stochastic global maximum principle for optimization with recursive utilities[J]. Probability, Uncertainty and Quantitative Risk, 2017, 2(1):1-20.
[19] WU Z, YU Z Y. Dynamic programming principle for one kind of stochastic recursive optimal control problem and Hamilton-Jacobi-Bellman equation[J]. SIAM Journal on Control and Optimization, 2008, 47(5):2616-2641.
[20] NIE T Y, SHI J T, WU Z. Connection between MP and DPP for stochastic recursive optimal control problems: viscosity solution framework in the general case[J]. SIAM Journal on Control and Optimization, 2017, 55(5):3258-3294.
[21] WONHAM W M. On the separation theorem of stochastic control[J]. SIAM Journal on Control, 1968, 6(2):312-326.
[22] LI X J, TANG S J. General necessary conditions for partially observed optimal stochastic controls[J]. Journal of Applied Probability, 1995, 32(4):1118-1137.
[23] TANG S J. The maximum principle for partially observed optimal control of stochastic differential equations[J]. SIAM Journal on Control and Optimization, 1998, 36(5):1596-1617.
[24] WANG G C, WU Z. Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems[J]. Journal of Mathematical Analysis and Applications, 2008, 342(2):1280-1296.
[25] WANG G C, ZHANG C H, ZHANG W H. Stochastic maximum principle for mean-field type optimal control under partial information[J]. IEEE Transactions on Automatic Control, 2014, 59(2):522-528.
[26] WANG G C, WU Z, XIONG J. An introduction to optimal control of FBSDE with incomplete information[M]. Cham: Springer, 2018.
[27] WU Z. A maximum principle for partially observed optimal control of forward-backward stochastic control systems[J]. Science China Information Sciences, 2010, 53(11):2205-2214.
[28] WANG G C, WU Z, XIONG J. Maximum principles for forward-backward stochastic control systems with correlated state and observation noises[J]. SIAM Journal on Control and Optimization, 2013, 51(1):491-524.
[29] WANG G C, WU Z, XIONG J. A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information[J]. IEEE Transactions on Automatic Control, 2015, 60(11):2904-2916.
[30] KOHLMANN M, ZHOU X Y. Relationship between backward stochastic differential equations and stochastic controls: a linear-quadratic approach[J]. SIAM Journal on Control and Optimization, 2000, 38(5):1392-1407.
[31] LIM A E B, ZHOU X Y. Linear-quadratic control of backward stochastic differential equations[J]. SIAM Journal on Control and Optimization, 2001, 40(2):450-474.
[32] HU M S, JI S L, XUE X L. A global stochastic maximum principle for fully coupled forward-backward stochastic systems[J]. SIAM Journal on Control and Optimization, 2018, 56(6):4309-4335.
[33] MOON J. The risk-sensitive maximum principle for controlled forward-backward stochastic differential equations[J]. Automatica, 2020, 120:109069.
[34] JI S L, LIU H. Maximum principle for stochastic optimal control problem of forward-backward stochastic difference systems[J/OL].(2021-12-29). International Journal of Control. https://arxiv.org/abs/1812.11283.
[35] LI R J, FU F Y. The maximum principle for partially observed optimal control problems of mean-field FBSDEs[J]. International Journal of Control, 2019, 92(10):2463-2472. doi:10.1080/00207179.2018.1441555.
[36] ZHANG S Q, LI X, XIONG J. A stochastic maximum principle for partially observed stochastic control systems with delay[J]. Systems & Control Letters, 2020, 146:1-7. doi:10.1016/j.sysconle.2020.104812.
[37] BENSOUSSAN A. Stochastic control of partially observable systems[M]. Cambridge: Cambridge University Press, 1992.
[38] HUANG J H, WANG G C, XIONG J. A maximum principle for partial information backward stochastic control problems with applications[J]. SIAM Journal on Control and Optimization, 2009, 48(4):2106-2117.
[39] LI N, WANG G C, WU Z. Linear-quadratic optimal control for time-delay stochastic system with recursive utility under full and partial information[J]. Automatica, 2020, 121:109169.
[40] BENSOUSSAN A, FENG X W, HUANG J H. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise[J]. Mathematical Control & Related Fields, 2021, 11(1):23-46.
[41] WANG G C, WANG W C, YAN Z G. Linear quadratic control of backward stochastic differential equation with partial information[J]. Applied Mathematics and Computation, 2021, 403:126164.
[1] 李永明,聂彩玲,刘超,郭建华. 负超可加阵列下非参数回归函数估计的相合性[J]. 《山东大学学报(理学版)》, 2018, 53(12): 69-74.
[2] 陈昊君,郑莹,马明,边莉娜,刘华. 自激滤过泊松过程的协方差[J]. 《山东大学学报(理学版)》, 2018, 53(12): 75-79.
[3] 周玉兰,李转,李晓慧. 连续时间Guichardet-Fock空间中修正随机梯度算子的性质[J]. 《山东大学学报(理学版)》, 2018, 53(12): 62-68.
[4] 李小娟,高强. 次线性期望框架下乘积空间的正则性[J]. 山东大学学报（理学版）, 2018, 53(4): 66-75.
[5] 马明,边莉娜,刘华. 基于事件点联合分布的自激滤过泊松过程的低阶矩[J]. 山东大学学报（理学版）, 2018, 53(4): 55-58.
[6] 肖新玲. 由马氏链驱动的正倒向随机微分方程[J]. 山东大学学报（理学版）, 2018, 53(4): 46-54.
[7] 陈丽,林玲. 具有时滞效应的股票期权定价[J]. 山东大学学报（理学版）, 2018, 53(4): 36-41.
[8] 张亚娟,吕艳. 带有变阻尼的随机振动方程的逼近[J]. 山东大学学报（理学版）, 2018, 53(4): 59-65.
[9] 崔静,梁秋菊. 分数布朗运动驱动的非局部随机积分微分系统的存在性与可控性[J]. 山东大学学报（理学版）, 2017, 52(12): 81-88.
[10] 黄爱玲,林帅. 局部量子Bernoulli噪声意义下的随机Schrödinger方程的有限维逼近[J]. 山东大学学报（理学版）, 2017, 52(12): 67-71.
[11] 荣文萍,崔静. 非Lipschitz条件下一类随机发展方程的μ-概几乎自守解[J]. 山东大学学报（理学版）, 2017, 52(10): 64-71.
[12] 冯德成,张潇,周霖. 弱鞅的一类极小值不等式[J]. 山东大学学报（理学版）, 2017, 52(8): 65-69.
[13] 张节松. 现代风险模型的扩散逼近与最优投资[J]. 山东大学学报（理学版）, 2017, 52(5): 49-57.
[14] 杨叙,李硕. 白噪声和泊松随机测度驱动的倒向重随机微分方程的比较定理[J]. 山东大学学报（理学版）, 2017, 52(4): 26-29.
[15] 张亚运,吴群英. ρ-混合序列的重对数律矩收敛的精确渐近性[J]. 山东大学学报（理学版）, 2017, 52(4): 13-20.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发