American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Semicontinuity of approximate solution mappings to generalized vector equilibrium problems
  • JIMO Home
  • This Issue
  • Next Article
    Stability analysis of a delayed social epidemics model with general contact rate and its optimal control
October  2016, 12(4): 1287-1301. doi: 10.3934/jimo.2016.12.1287

A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises

Haiyan Zhang 1,

1. 

College of Sciences, Shandong Jiaotong University, Jinan 250023, China

Received  March 2015 Revised  June 2015 Published  January 2016

This paper is concerned with a mean-field type optimal control problem, whose new features are that the state $x^v_t$ is partially observed by a noisy process $y(t)$, and the control problem is time inconsistent in the sense that Bellman optimality principle does not work. A necessary condition for optimality is derived by convex variation, dual technique and backward stochastic differential equations (BSDEs). A linear-quadratic (LQ) optimal control example is studied, and the optimal solution is obtained by the optimal filtering for BSDEs and the necessary condition.
Keywords: filtering., LQ control, necessary condition, correlated state and observation noises, Mean-field control.
Mathematics Subject Classification: Primary: 93E11, 93E20; Secondary: 60H1.
Citation: Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287
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.  Google Scholar

[2]

A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, 1992. doi: 10.1017/CBO9780511526503.  Google Scholar

[3]

R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), 3133-3154. doi: 10.1016/j.spa.2009.05.002.  Google Scholar

[4]

X. Cui, X. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Trans. Automat. Control, 59 (2014), 1833-1844. doi: 10.1109/TAC.2014.2311875.  Google Scholar

[5]

R. Elliott, X. Li and Y. Ni, Discrete time mean-field stochastic linear quadratic optimal control problems, Automatica, 49 (2013), 3222-3233. doi: 10.1016/j.automatica.2013.08.017.  Google Scholar

[6]

M. Hafayed, A mean-field maximum principle for optimal control of forward-backward stochastic differential equations with Poisson jump processes, Int. J. Dynam. Control, 1 (2013), 300-315. doi: 10.1007/s40435-013-0027-8.  Google Scholar

[7]

M. Hafayed, A mean-field necessary and sufficient conditions for optimal singular stochastic control, Commun. Math. Stat., 1 (2013), 417-435. doi: 10.1007/s40304-014-0023-0.  Google Scholar

[8]

M. Hafayed, Singular mean-field optimal control for forward-backward stochastic systems and applications to finance, Int. J. Dynam. Control, 2 (2014), 542-554. doi: 10.1007/s40435-014-0080-y.  Google Scholar

[9]

M. Hafayed, A. Abba and S. Abbas, On mean-field stochastic maximum principle for near optimal controls for poisson jump diffusion with applications, Int. J. Dynam. Control, 2 (2014), 262-284. doi: 10.1007/s40435-013-0040-y.  Google Scholar

[10]

M. Hafayed and S. Abbas, On near-optimal mean-field stochastic singular controls: Necessary and sufficient conditions for near-optimality, J. Optim. Theory Appl., 160 (2014), 778-808. doi: 10.1007/s10957-013-0361-1.  Google Scholar

[11]

J. Huang, X. Li and J. Yong, A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon, Math. Control Relat. Fields, 5 (2015), 97-139. doi: 10.3934/mcrf.2015.5.97.  Google Scholar

[12]

J. Huang, G. Wang and Z. Wu, Optimal premium policy of an insurance firm: Full and partial information, Insurance: Math. Econ., 47 (2010), 208-215. doi: 10.1016/j.insmatheco.2010.04.007.  Google Scholar

[13]

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

[14]

Y. Ni, J. Zhang and X. Li, Indefinite mean-field stochastic linear-quadratic optimal control, IEEE Trans. Automat. Control, 60 (2015), 1786-1800. doi: 10.1109/TAC.2014.2385253.  Google Scholar

[15]

G. Wang and Z. Wu, Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, J. Math. Anal. Appl., 342 (2008), 1280-1296. doi: 10.1016/j.jmaa.2007.12.072.  Google Scholar

[16]

G. Wang, Z. Wu and J. Xiong, Maximum principle for forward-backward stochastic control systems with corrected state and observation noises, SIAM J. Control Optim., 51 (2013), 491-524. doi: 10.1137/110846920.  Google Scholar

[17]

G. Wang, Z. Wu and C. Zhang, Maximum principles for partially observed mean-field stochastic systems with applications to financial engineering, Proceedings of the 33rd Chinese Control Conference, July 28-30, 2014, Nanjing, China, 5357-5362. doi: 10.1109/ChiCC.2014.6895853.  Google Scholar

[18]

G. Wang, C. Zhang and W. Zhang, Stochastic maximum principle for mean-field type optimal control under partial information, IEEE Trans. Automat. Control, 59 (2014), 522-528. doi: 10.1109/TAC.2013.2273265.  Google Scholar

[19]

W. M. Wonham, On the separation theorem of stochastic control, SIAM J. Control, 6 (1968), 312-326. doi: 10.1137/0306023.  Google Scholar

[20]

H. Xiao and G. Wang, The filtering equations of forward-backward stochastic systems with random jumps and applications to partial information stochastic optimal control, Stoch. Anal. Appl., 28 (2010), 1003-1019. doi: 10.1080/07362994.2010.515480.  Google Scholar

[21]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, 2008.  Google Scholar

[22]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim., 51 (2013), 2809-2838. doi: 10.1137/120892477.  Google Scholar

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.  Google Scholar

[2]

A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, 1992. doi: 10.1017/CBO9780511526503.  Google Scholar

[3]

R. Buckdahn, J. Li and S. Peng, Mean-field backward stochastic differential equations and related partial differential equations, Stochastic Process. Appl., 119 (2009), 3133-3154. doi: 10.1016/j.spa.2009.05.002.  Google Scholar

[4]

X. Cui, X. Li and D. Li, Unified framework of mean-field formulations for optimal multi-period mean-variance portfolio selection, IEEE Trans. Automat. Control, 59 (2014), 1833-1844. doi: 10.1109/TAC.2014.2311875.  Google Scholar

[5]

R. Elliott, X. Li and Y. Ni, Discrete time mean-field stochastic linear quadratic optimal control problems, Automatica, 49 (2013), 3222-3233. doi: 10.1016/j.automatica.2013.08.017.  Google Scholar

[6]

M. Hafayed, A mean-field maximum principle for optimal control of forward-backward stochastic differential equations with Poisson jump processes, Int. J. Dynam. Control, 1 (2013), 300-315. doi: 10.1007/s40435-013-0027-8.  Google Scholar

[7]

M. Hafayed, A mean-field necessary and sufficient conditions for optimal singular stochastic control, Commun. Math. Stat., 1 (2013), 417-435. doi: 10.1007/s40304-014-0023-0.  Google Scholar

[8]

M. Hafayed, Singular mean-field optimal control for forward-backward stochastic systems and applications to finance, Int. J. Dynam. Control, 2 (2014), 542-554. doi: 10.1007/s40435-014-0080-y.  Google Scholar

[9]

M. Hafayed, A. Abba and S. Abbas, On mean-field stochastic maximum principle for near optimal controls for poisson jump diffusion with applications, Int. J. Dynam. Control, 2 (2014), 262-284. doi: 10.1007/s40435-013-0040-y.  Google Scholar

[10]

M. Hafayed and S. Abbas, On near-optimal mean-field stochastic singular controls: Necessary and sufficient conditions for near-optimality, J. Optim. Theory Appl., 160 (2014), 778-808. doi: 10.1007/s10957-013-0361-1.  Google Scholar

[11]

J. Huang, X. Li and J. Yong, A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon, Math. Control Relat. Fields, 5 (2015), 97-139. doi: 10.3934/mcrf.2015.5.97.  Google Scholar

[12]

J. Huang, G. Wang and Z. Wu, Optimal premium policy of an insurance firm: Full and partial information, Insurance: Math. Econ., 47 (2010), 208-215. doi: 10.1016/j.insmatheco.2010.04.007.  Google Scholar

[13]

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

[14]

Y. Ni, J. Zhang and X. Li, Indefinite mean-field stochastic linear-quadratic optimal control, IEEE Trans. Automat. Control, 60 (2015), 1786-1800. doi: 10.1109/TAC.2014.2385253.  Google Scholar

[15]

G. Wang and Z. Wu, Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, J. Math. Anal. Appl., 342 (2008), 1280-1296. doi: 10.1016/j.jmaa.2007.12.072.  Google Scholar

[16]

G. Wang, Z. Wu and J. Xiong, Maximum principle for forward-backward stochastic control systems with corrected state and observation noises, SIAM J. Control Optim., 51 (2013), 491-524. doi: 10.1137/110846920.  Google Scholar

[17]

G. Wang, Z. Wu and C. Zhang, Maximum principles for partially observed mean-field stochastic systems with applications to financial engineering, Proceedings of the 33rd Chinese Control Conference, July 28-30, 2014, Nanjing, China, 5357-5362. doi: 10.1109/ChiCC.2014.6895853.  Google Scholar

[18]

G. Wang, C. Zhang and W. Zhang, Stochastic maximum principle for mean-field type optimal control under partial information, IEEE Trans. Automat. Control, 59 (2014), 522-528. doi: 10.1109/TAC.2013.2273265.  Google Scholar

[19]

W. M. Wonham, On the separation theorem of stochastic control, SIAM J. Control, 6 (1968), 312-326. doi: 10.1137/0306023.  Google Scholar

[20]

H. Xiao and G. Wang, The filtering equations of forward-backward stochastic systems with random jumps and applications to partial information stochastic optimal control, Stoch. Anal. Appl., 28 (2010), 1003-1019. doi: 10.1080/07362994.2010.515480.  Google Scholar

[21]

J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, 2008.  Google Scholar

[22]

J. Yong, Linear-quadratic optimal control problems for mean-field stochastic differential equations, SIAM J. Control Optim., 51 (2013), 2809-2838. doi: 10.1137/120892477.  Google Scholar

[1]

Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control & Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026
[2]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[3]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[4]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026
[5]

Xun Li, Jingrui Sun, Jiongmin Yong. Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 2-. doi: 10.1186/s41546-016-0002-3
[6]

Liangquan Zhang, Qing Zhou, Juan Yang. Necessary condition for optimal control of doubly stochastic systems. Mathematical Control & Related Fields, 2020, 10 (2) : 379-403. doi: 10.3934/mcrf.2020002
[7]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic & Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
[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 & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035
[9]

Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021074
[10]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004
[11]

Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022
[12]

Michael Herty, Lorenzo Pareschi, Sonja Steffensen. Mean--field control and Riccati equations. Networks & Heterogeneous Media, 2015, 10 (3) : 699-715. doi: 10.3934/nhm.2015.10.699
[13]

Charlotte Beauthier, Joseph J. Winkin, Denis Dochain. Input/state invariant LQ-optimal control: Application to competitive coexistence in a chemostat. Evolution Equations & Control Theory, 2015, 4 (2) : 143-158. doi: 10.3934/eect.2015.4.143
[14]

Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082
[15]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006
[16]

Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299
[17]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
[18]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011
[19]

Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096
[20]

Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (135)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences