JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2021, Vol. 56 ›› Issue (10): 1-10.doi: 10.6040/j.issn.1671-9352.9.2021.012

   

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)

905

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

CLC Number: 

  • 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] LI Yong-ming, NIE Cai-ling, LIU Chao, GUO Jian-hua. Consistency of estimator of nonparametric regression function for arrays of rowwise NSD [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(12): 69-74.
[2] CHEN Hao-jun, ZHENG Ying, MA Ming, BIAN Li-na, LIU Hua. Covariance of self-exciting filtered Poisson process [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(12): 75-79.
[3] ZHOU Yu-lan, LI Zhuan, LI Xiao-hui. Properties of modified stochastic gradient operators in continuous-time Guichardet-Fock space [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(12): 62-68.
[4] LI Xiao-juan, GAO Qiang. Regularity for product space under sublinear expectation framework [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 66-75.
[5] MA Ming, BIAN Li-na, LIU Hua. Low order moments of self-excited filtered Poisson processes based on joint distribution of event points [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 55-58.
[6] XIAO Xin-ling. Forward-backward stochastic differential equations on Markov chains [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 46-54.
[7] CHEN Li, LIN Ling. Stock option pricing with time delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 36-41.
[8] ZHANG Ya-juan, LYU Yan. Approximation of stochastic vibration equations with variable damping [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(4): 59-65.
[9] CUI Jing, LIANG Qiu-ju. Existence and controllability of nonlocal stochastcic integro-differential equations driven by fractional Brownian motion [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 81-88.
[10] HUANG Ai-ling, LIN Shuai. Finite dimensional approximation of linear stochastic Schrödinger equation in terms of localization of quantum Bernoulli noises [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 67-71.
[11] RONG Wen-ping, CUI Jing. μ-pseudo almost automorphic solutions for a class of stochastic evolution equations under non-Lipschitz conditions [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 64-71.
[12] FENG De-cheng, ZHANG Xiao, ZHOU Lin. A class of minimal inequalities for demimartingales [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 65-69.
[13] ZHANG Jie-song. Diffusion approximation and optimal investment for modern risk model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(5): 49-57.
[14] YANG Xu, LI Shuo. Comparison theorem for backward doubly stochastic differential equations driven by white noises and Poisson random measures [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 26-29.
[15] ZHANG Ya-yun, WU Qun-ying. Precise asymptotics in the law of iterated logarithm for the moment convergence of ρ-mixing sequences [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(4): 13-20.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd