logo

SCIENCE CHINA Information Sciences, Volume 63 , Issue 9 : 192204(2020) https://doi.org/10.1007/s11432-019-1473-3

A partial information linear-quadratic optimal control problem of backward stochastic differential equation with its applications

More info
  • ReceivedMay 30, 2019
  • AcceptedJul 10, 2019
  • PublishedJul 28, 2020

Abstract

In this paper, we investigate a kind of partial information linear-quadratic optimal control problem driven by a backward stochastic differential equation, where the state equation and the cost functional contain diffusion terms. Using maximum principle, we derive the corresponding Hamiltonian system, which is a conditional mean-field forward-backward stochastic differential equation. By the backward separation approach and the filtering technique, we get two Riccati equations, and a backward and a forward optimal filtering equations. Then a feedback form of optimal control is obtained.We also extend thecontrol problem to the case of mean-field backward stochastic differential equation under partial information.A corresponding feedback form of optimal control is also obtained.


Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 11371228, 61821004, 61633015). The authors would like to thank the anonymous referees for their valuable comments, which led to a much better version of this paper.


References

[1] Dokuchaev N, Zhou X Y. Stochastic Controls with Terminal Contingent Conditions. J Math Anal Appl, 1999, 238: 143-165 CrossRef Google Scholar

[2] 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 CrossRef Google Scholar

[3] Lim A E B, Zhou X Y. Linear-Quadratic Control of Backward Stochastic Differential Equations. SIAM J Control Optim, 2001, 40: 450-474 CrossRef Google Scholar

[4] Yu Z Y. Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J Control, 2012, 14: 173-185 CrossRef Google Scholar

[5] Wu S, Wang G C. Optimal control problem of backward stochastic differential delay equation under partial information. Syst Control Lett, 2015, 82: 71-78 CrossRef Google Scholar

[6] Zhang S Q, Xiong J, Liu X D. Stochastic maximum principle for partially observed forward-backward stochastic differential equations with jumps and regime switching. Sci China Inf Sci, 2018, 61: 070211 CrossRef Google Scholar

[7] Kac M. Foundations of kinetic theory. In: Proceedings of the 3th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, 1956. 171--197. Google Scholar

[8] Yong J M. Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations. SIAM J Control Optim, 2013, 51: 2809-2838 CrossRef Google Scholar

[9] Li X, Sun J R, Xiong J. Linear Quadratic Optimal Control Problems for Mean-Field Backward Stochastic Differential Equations. Appl Math Optim, 2019, 80: 223-250 CrossRef Google Scholar

[10] Douissi S, Wen J Q, Shi Y F. Mean-field anticipated BSDEs driven by fractional Brownian motion and related stochastic control problem. Appl Math Computation, 2019, 355: 282-298 CrossRef Google Scholar

[11] Ma H P. Infinite horizon optimal control of mean-field forward-backward delayed systems with Poisson jumps. Eur J Control, 2019, 46: 14-22 CrossRef Google Scholar

[12] Huang J H, Wang G C, Xiong J. A Maximum Principle for Partial Information Backward Stochastic Control Problems with Applications. SIAM J Control Optim, 2009, 48: 2106-2117 CrossRef Google Scholar

[13] Wang G C, Wu Z, Xiong J. A Linear-Quadratic Optimal Control Problem of Forward-Backward Stochastic Differential Equations With Partial Information. IEEE Trans Automat Contr, 2015, 60: 2904-2916 CrossRef Google Scholar

[14] Huang J H, Wang S J, Wu Z. Backward Mean-Field Linear-Quadratic-Gaussian (LQG) Games: Full and Partial Information. IEEE Trans Automat Contr, 2016, 61: 3784-3796 CrossRef Google Scholar

[15] Wang G C, Xiao H, Xiong J. A kind of LQ non-zero sum differential game of backward stochastic differential equation with asymmetric information. Automatica, 2018, 97: 346-352 CrossRef Google Scholar

[16] Wu Z. A maximum principle for partially observed optimal control of forward-backward stochastic control systems. Sci China Inf Sci, 2010, 53: 2205-2214 CrossRef Google Scholar

[17] Xiong J. An Introduction to Stochastic Filtering Theory. London: Oxford University Press, 2008. Google Scholar

[18] Yu Z Y, Ji S L. Linear-quadratic nonzero-sum differential game of backward stochastic differential equations. In: Proceedings of the 27th Chinese Control Conference, Kunming, 2008. 562--566. Google Scholar

[19] Buckdahn R, Li J, Peng S G. Mean-field backward stochastic differential equations and related partial differential equations. Stochastic Processes their Appl, 2009, 119: 3133-3154 CrossRef Google Scholar

[20] 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 CrossRef Google Scholar

[21] Huang P Y, Wang G C, Zhang H J. An asymmetric information non-zero sum differential game of mean-field backward stochastic differential equation with applications. Adv Differ Equ, 2019, 2019(1): 236 CrossRef Google Scholar

Copyright 2020  CHINA SCIENCE PUBLISHING & MEDIA LTD.  中国科技出版传媒股份有限公司  版权所有

京ICP备14028887号-23       京公网安备11010102003388号