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

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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

Abstract
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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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2014), 28. 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. doi: 10.1109/TAC.2013.2273265.
[19]	W. M. Wonham, On the separation theorem of stochastic control,, SIAM J. Control, 6 (1968), 312. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2014), 28. 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. doi: 10.1109/TAC.2013.2273265.
[19]	W. M. Wonham, On the separation theorem of stochastic control,, SIAM J. Control, 6 (1968), 312. 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. 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. doi: 10.1137/120892477.

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