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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

 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.
Citation: Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial and 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. [2] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, 1992. doi: 10.1017/CBO9780511526503. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] W. M. Wonham, On the separation theorem of stochastic control, SIAM J. Control, 6 (1968), 312-326. doi: 10.1137/0306023. [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. [21] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, 2008. [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.

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##### 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, Stochastic Control of Partially Observable Systems, Cambridge University Press, 1992. doi: 10.1017/CBO9780511526503. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [19] W. M. Wonham, On the separation theorem of stochastic control, SIAM J. Control, 6 (1968), 312-326. doi: 10.1137/0306023. [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. [21] J. Xiong, An Introduction to Stochastic Filtering Theory, Oxford University Press, 2008. [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.
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