• Previous Article
    Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability
  • MCRF Home
  • This Issue
  • Next Article
    Barrier Lyapunov functions-based adaptive neural tracking control for non-strict feedback stochastic nonlinear systems with full-state constraints: A command filter approach
doi: 10.3934/mcrf.2022012
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A general maximum principle for partially observed mean-field stochastic system with random jumps in progressive structure

1. 

Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan Shandong 250100, China

2. 

School of Mathematics, Shandong University, Jinan Shandong 250100, China

* Corresponding author: Zhen Wu

Received  September 2021 Revised  January 2022 Early access March 2022

Fund Project: This work was supported by the Natural Science Foundation of China under Grant (11831010, 61961160732), Shandong Provincial Natural Science Foundation (ZR2019ZD42) and The Taishan Scholars Climbing Program of Shandong (TSPD20210302)

We study the progressive optimal control for partially observed stochastic system of mean-field type with random jumps. The cost function and the observation are also of mean-field type. The control is allowed to enter the diffusion, jump coefficient and the observation. The control domain need not be convex. We obtain the maximum principle for the partially observable progressive optimal control by a special spike variation. The maximum principle in the progressive structure is different from the classical case.

Citation: Tian Chen, Zhen Wu. A general maximum principle for partially observed mean-field stochastic system with random jumps in progressive structure. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022012
References:
[1]

D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8.

[2]

A. Bensoussan, Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions, Stochastics, 9 (1983), 169-222.  doi: 10.1080/17442508308833253.

[3]

J. M. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev., 20 (1978), 62-78.  doi: 10.1137/1020004.

[4]

R. BuckdahnB. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Appl. Math. Optim., 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y.

[5]

U. G. Haussmann, The maximum principle for optimal control of diffusions with partial information, SIAM J. Control Optim., 25 (1987), 341-361.  doi: 10.1137/0325021.

[6]

M. Hu, Stochastic global maximum principle for optimization with recursive utilities, Probab. Uncertain. Quant. Risk, 2 (2017), 1-20.  doi: 10.1186/s41546-017-0014-7.

[7]

R. Li and F. Fu, The maximum principle for partially observed optimal control problems of mean-field FBSDEs, Int. J. Control, 92 (2019), 2463-2472.  doi: 10.1080/00207179.2018.1441555.

[8]

X. Li and S. Tang, General necessary conditions for partially observed optimal stochastic controls, J. Appl. Probab., 32 (1995), 1118-1137.  doi: 10.2307/3215225.

[9]

T. Meyer-BrandisB. Øksendal and X. Y. Zhou, A mean-field stochastic maximum principle via Malliavin calculus, Stochastics, 84 (2012), 643-666.  doi: 10.1080/17442508.2011.651619.

[10]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.

[11]

Y. ShenQ. Meng and P. Shi, Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance, Automatica J. IFAC, 50 (2014), 1565-1579.  doi: 10.1016/j.automatica.2014.03.021.

[12]

Y. Shen and T. K. Siu, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Anal.-Theory Methods Appl., 86 (2013), 58-73.  doi: 10.1016/j.na.2013.02.029.

[13]

Y. SongS. Tang and Z. Wu, The maximum principle for progressive optimal stochastic control problems with random jumps, SIAM J. Control Optim., 58 (2020), 2171-2187.  doi: 10.1137/19M1292308.

[14]

J. SunH. Wang and Z. Wu, Mean-field linear-quadratic stochastic differential games, J. Differ. Equ., 296 (2021), 299-334.  doi: 10.1016/j.jde.2021.06.004.

[15]

Z. Sun and O. Menoukeu-Pamen, The maximum principles for partially observed risk-sensitive optimal controls of Markov regime-switching jump-diffusion system, Stoch. Anal. Appl., 36 (2018), 782-811.  doi: 10.1080/07362994.2018.1465824.

[16]

S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations, SIAM J. Control Optim., 36 (1998), 1596-1617.  doi: 10.1137/S0363012996313100.

[17]

S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32 (1994), 1447-1475.  doi: 10.1137/S0363012992233858.

[18]

G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information, IEEE Trans. Autom. Control, 54 (2009), 1230-1242.  doi: 10.1109/TAC.2009.2019794.

[19]

Z. Wu, A maximum principle for partially observed optimal control of forward-backward stochastic control systems, Sci. China-Inf. Sci., 53 (2010), 2205-2214.  doi: 10.1007/s11432-010-4094-6.

[20]

Z. Wu, A general maximum principle for optimal control of forward-backward stochastic systems, Automatica, 49 (2013), 1473-1480.  doi: 10.1016/j.automatica.2013.02.005.

[21]

W. Xu, Stochastic maximum principle for optimal control problem of forward and backward system, Anziam J., 37 (1995), 172-185.  doi: 10.1017/S0334270000007645.

[22]

X. ZhangZ. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM J. Control Optim., 56 (2018), 2563-2592.  doi: 10.1137/17M112395X.

[23]

Q. ZhouY. Ren and W. Wu, On optimal mean-field control problem of mean-field forward-backward stochastic system with jumps under partial information, J. Syst. Sci. Complex., 30 (2017), 828-856.  doi: 10.1007/s11424-016-5237-7.

[24]

X. Y. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1993), 1462-1478.  doi: 10.1137/0331068.

show all references

References:
[1]

D. Andersson and B. Djehiche, A maximum principle for SDEs of mean-field type, Appl. Math. Optim., 63 (2011), 341-356.  doi: 10.1007/s00245-010-9123-8.

[2]

A. Bensoussan, Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions, Stochastics, 9 (1983), 169-222.  doi: 10.1080/17442508308833253.

[3]

J. M. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Rev., 20 (1978), 62-78.  doi: 10.1137/1020004.

[4]

R. BuckdahnB. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type, Appl. Math. Optim., 64 (2011), 197-216.  doi: 10.1007/s00245-011-9136-y.

[5]

U. G. Haussmann, The maximum principle for optimal control of diffusions with partial information, SIAM J. Control Optim., 25 (1987), 341-361.  doi: 10.1137/0325021.

[6]

M. Hu, Stochastic global maximum principle for optimization with recursive utilities, Probab. Uncertain. Quant. Risk, 2 (2017), 1-20.  doi: 10.1186/s41546-017-0014-7.

[7]

R. Li and F. Fu, The maximum principle for partially observed optimal control problems of mean-field FBSDEs, Int. J. Control, 92 (2019), 2463-2472.  doi: 10.1080/00207179.2018.1441555.

[8]

X. Li and S. Tang, General necessary conditions for partially observed optimal stochastic controls, J. Appl. Probab., 32 (1995), 1118-1137.  doi: 10.2307/3215225.

[9]

T. Meyer-BrandisB. Øksendal and X. Y. Zhou, A mean-field stochastic maximum principle via Malliavin calculus, Stochastics, 84 (2012), 643-666.  doi: 10.1080/17442508.2011.651619.

[10]

S. Peng, A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28 (1990), 966-979.  doi: 10.1137/0328054.

[11]

Y. ShenQ. Meng and P. Shi, Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance, Automatica J. IFAC, 50 (2014), 1565-1579.  doi: 10.1016/j.automatica.2014.03.021.

[12]

Y. Shen and T. K. Siu, The maximum principle for a jump-diffusion mean-field model and its application to the mean-variance problem, Nonlinear Anal.-Theory Methods Appl., 86 (2013), 58-73.  doi: 10.1016/j.na.2013.02.029.

[13]

Y. SongS. Tang and Z. Wu, The maximum principle for progressive optimal stochastic control problems with random jumps, SIAM J. Control Optim., 58 (2020), 2171-2187.  doi: 10.1137/19M1292308.

[14]

J. SunH. Wang and Z. Wu, Mean-field linear-quadratic stochastic differential games, J. Differ. Equ., 296 (2021), 299-334.  doi: 10.1016/j.jde.2021.06.004.

[15]

Z. Sun and O. Menoukeu-Pamen, The maximum principles for partially observed risk-sensitive optimal controls of Markov regime-switching jump-diffusion system, Stoch. Anal. Appl., 36 (2018), 782-811.  doi: 10.1080/07362994.2018.1465824.

[16]

S. Tang, The maximum principle for partially observed optimal control of stochastic differential equations, SIAM J. Control Optim., 36 (1998), 1596-1617.  doi: 10.1137/S0363012996313100.

[17]

S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32 (1994), 1447-1475.  doi: 10.1137/S0363012992233858.

[18]

G. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information, IEEE Trans. Autom. Control, 54 (2009), 1230-1242.  doi: 10.1109/TAC.2009.2019794.

[19]

Z. Wu, A maximum principle for partially observed optimal control of forward-backward stochastic control systems, Sci. China-Inf. Sci., 53 (2010), 2205-2214.  doi: 10.1007/s11432-010-4094-6.

[20]

Z. Wu, A general maximum principle for optimal control of forward-backward stochastic systems, Automatica, 49 (2013), 1473-1480.  doi: 10.1016/j.automatica.2013.02.005.

[21]

W. Xu, Stochastic maximum principle for optimal control problem of forward and backward system, Anziam J., 37 (1995), 172-185.  doi: 10.1017/S0334270000007645.

[22]

X. ZhangZ. Sun and J. Xiong, A general stochastic maximum principle for a Markov regime switching jump-diffusion model of mean-field type, SIAM J. Control Optim., 56 (2018), 2563-2592.  doi: 10.1137/17M112395X.

[23]

Q. ZhouY. Ren and W. Wu, On optimal mean-field control problem of mean-field forward-backward stochastic system with jumps under partial information, J. Syst. Sci. Complex., 30 (2017), 828-856.  doi: 10.1007/s11424-016-5237-7.

[24]

X. Y. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1993), 1462-1478.  doi: 10.1137/0331068.

[1]

Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial and Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067

[2]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control and Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022

[3]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100

[4]

Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499

[5]

Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control and Related Fields, 2022, 12 (2) : 475-493. doi: 10.3934/mcrf.2021031

[6]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control and Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018

[7]

Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7

[8]

Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations and Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035

[9]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[10]

Jian Song, Meng Wang. Stochastic maximum principle for systems driven by local martingales with spatial parameters. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 213-236. doi: 10.3934/puqr.2021011

[11]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control and Related Fields, 2021, 11 (4) : 829-855. doi: 10.3934/mcrf.2020048

[12]

Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control and Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195

[13]

H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77

[14]

H. O. Fattorini. The maximum principle in infinite dimension. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557

[15]

Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221

[16]

Dejian Chang, Zhen Wu. Stochastic maximum principle for non-zero sum differential games of FBSDEs with impulse controls and its application to finance. Journal of Industrial and Management Optimization, 2015, 11 (1) : 27-40. doi: 10.3934/jimo.2015.11.27

[17]

Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174

[18]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[19]

Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control and Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021

[20]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110

2021 Impact Factor: 1.141

Metrics

  • PDF downloads (227)
  • HTML views (140)
  • Cited by (0)

Other articles
by authors

[Back to Top]