×

Linear quadratic control of backward stochastic differential equation with partial information. (English) Zbl 1510.93372

Summary: In this paper, we study an optimal control problem of linear backward stochastic differential equation (BSDE) with quadratic cost functional under partial information. This problem is solved completely and explicitly by using a stochastic maximum principle and a decoupling technique. In terms of the maximum principle, a stochastic Hamiltonian system, which is a forward-backward stochastic differential equation (FBSDE) with filtering, is obtained. By decoupling the stochastic Hamiltonian system, three Riccati equations, a BSDE with filtering, and a stochastic differential equation (SDE) with filtering are derived. We then get a feedback representation of optimal control. An explicit formula for the corresponding optimal cost is also established. As illustrative examples, we consider two special scalar-valued control problems and give some numerical simulations.

MSC:

93E20 Optimal stochastic control
49N10 Linear-quadratic optimal control problems
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bismut, J. M., An introductory approach to duality in optimal stochastic control, SIAM Rev., 20, 1, 62-78 (1978) · Zbl 0378.93049
[2] Pardoux, E.; Peng, S., Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14, 1, 55-61 (1990) · Zbl 0692.93064
[3] Karoui, N. E.; Peng, S.; Quenez, M. C., Backward stochastic differential equations in finance, Math. Financ., 7, 1-71 (1997) · Zbl 0884.90035
[4] Ma, J.; Yong, J., Forward-backward stochastic differential equations and their applications, Lecture Notes in Math (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0927.60004
[5] Kohlmann, M.; Zhou, X., Relationship between backward stochastic differential equations and stochastic controls: a linear-quadratic approach, SIAM J. Control Optim., 38, 5, 1392-1407 (2000) · Zbl 0960.60052
[6] Peng, S., Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim., 27, 2, 125-144 (1993) · Zbl 0769.60054
[7] Dokuchaev, N. G.; Zhou, X., Stochastic controls with terminal contingent conditions, J. Math. Anal. Appl., 238, 143-165 (1999) · Zbl 0937.93053
[8] Yan, Z.; Park, J. H.; Zhang, W., Finite-time guaranteed cost control for Itô stochastic markovian jump systems with incomplete transition rates, Int. J. Robust Nonlinear Control, 27, 1, 66-83 (2017) · Zbl 1353.93119
[9] Yan, Z.; Zhang, W.; Zhang, G., Finite-time stability and stabilization of Itô stochastic systems with markovian switching: mode-dependent parameter approach, IEEE Trans. Autom. Control, 60, 9, 2428-2433 (2015) · Zbl 1360.93757
[10] Yan, Z.; Zhang, G.; Zhang, W., Finite-time stability and stabilization of linear Itô stochastic systems with state and control-dependent noise, Asian J. Control, 15, 1, 270-281 (2013) · Zbl 1327.93393
[11] Xia, J.; Li, B.; Su, S.; Sun, W.; Shen, H., Finite-time command filtered event-triggered adaptive fuzzy tracking control for stochastic nonlinear systems, IEEE Trans. Fuzzy Syst. (2020)
[12] Yong, J.; Zhou, X., Stochastic Controls: Hamiltonian Systems and HJB Equations (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0943.93002
[13] Lim, A. E.B.; Zhou, X., Linear-quadratic control of backward stochastic differential equations, SIAM J. Control Optim., 40, 2, 450-474 (2001) · Zbl 0995.93074
[14] Li, X.; Sun, J.; Xiong, J., Linear quadratic optimal control problems for mean-field backward stochastic differential equations, Appl. Math. Optim., 80, 223-250 (2019) · Zbl 1428.49037
[15] Huang, J.; Wang, S.; Wu, Z., Backward mean-field linear-quadratic-gaussian (LQG) games: full and partial information, IEEE Trans. Autom. Control, 60, 12, 3784-3796 (2016) · Zbl 1359.91009
[16] Du, K.; Huang, J.; Wu, Z., Linear quadratic mean-field-game of backward stochastic differential systems, Math. Control Relat. Fields, 8, 653-678 (2018) · Zbl 1416.93198
[17] Du, K.; Wu, Z., Linear-quadratic Stackelberg game for mean-field backward stochastic differential system and application, Math. Probl. Eng., 17, 1-17 (2019) · Zbl 1435.91023
[18] Hu, Y.; Øksendal, B., Partial information linear quadratic control for jump diffusions, SIAM J. Control Optim., 47, 4, 1744-1761 (2008) · Zbl 1165.93037
[19] Wu, Z., A maximum principle for partially observed optimal control of forward-backward stochastic control systems, Sci. China Inf. Sci., 53, 11, 2205-2214 (2010) · Zbl 1227.93116
[20] Huang, J.; Wang, G.; Xiong, J., A maximum principle for partial information backward stochastic control problems with applications, SIAM J. Control Optim., 48, 4, 2106-2117 (2009) · Zbl 1203.49037
[21] Wang, G.; Wu, Z.; Xiong, J., A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information, IEEE Trans. Autom. Control, 60, 11, 2904-2916 (2015) · Zbl 1360.93787
[22] Wang, G.; Xiao, H.; Xing, G., An optimal control problem for mean-field forward-backward stochastic differential equation with noisy observation, Automatica, 86, 104-109 (2017) · Zbl 1375.93143
[23] Wang, G.; Xiao, H.; Xiong, J., A kind of LQ non-zero sum differential game of backward stochastic differential equation with asymmetric information, Automatica, 97, 346-352 (2018) · Zbl 1420.91022
[24] Wu, Z.; Zhuang, Y., Linear-quadratic partially observed forward-backward stochastic differential games and its application in finance, Appl. Math. Comput., 321, 577-592 (2018) · Zbl 1427.91028
[25] Huang, P.; Wang, G.; Zhang, H., A partial information linear-quadratic optimal control problem of backward stochastic differential equation with its applications, Sci. China Inf. Sci., 63, 9, 1-14 (2020)
[26] Øksendal, B., Stochastic Differential Equations: an Introduction with Applications (2005), Springer-Verlag: Springer-Verlag New York
[27] Wang, G.; Wu, Z.; Xiong, J., An Introduction to Optimal Control of FBSDE with Incomplete Information (2018), Springer-Verlag: Springer-Verlag New York · Zbl 1400.49001
[28] Xiong, J., An Introduction to Stochastic Filtering Theory (2008), Oxford University Press: Oxford University Press London · Zbl 1144.93003
[29] Liptser, R. S.; Shiryayev, A. N., Statistics of Random Processes (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0364.60004
[30] Ma, J.; Protter, P.; Martłn, J. S.; Torres, S., Numerical method for backward stochastic differential equations, Ann. Appl. Probab., 12, 302-316 (2002) · Zbl 1017.60074
[31] Peng, S.; Xu, M., Numerical algorithms for backward stochastic differential equations with 1-d Brownian motion: convergence and simulations, ESAIM Math. Model. Numer. Anal., 45, 335-360 (2011) · Zbl 1269.65008
[32] Zhao, W.; Chen, L.; Peng, S., A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28, 4, 1563-1581 (2006) · Zbl 1121.60072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.