Abstract
In this paper, we study the mean-field-type partial information stochastic optimal control problem, where the system is governed by a controlled stochastic differential equation, driven by the Teugels martingales associated with some Lévy processes and an independent Brownian motion. We derive necessary and sufficient conditions of the optimal control for these mean-field models in the form of a maximum principle. The control domain is assumed to be convex. As an application, the partial information linear quadratic control problem of the mean-field type is discussed.
Similar content being viewed by others
References
Meng, Q.X., Tang, M.N.: Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes. Sci. China Ser. F Inf. Sci. 52(11), 1982–1992 (2009)
Meng, Q.X., Zhang, F.,Tang, M.N.: Maximum principle for backward stochastic systems associated with Lévy processes under partial information, In: Proceedings of the 31 st Chinese control conference, July 25-27, Hefei, China (2012)
Mitsui, K., Tabata, M.: A stochastic linear quadratic problem with Lévy, process and its application to finance. Stoch. Process. Appl. 118, 120–152 (2008)
Tang, H., Wu, Z.: Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy processes. J. Syst. Sci. Complex. 22, 122–136 (2009)
Tang, M., Zhang, Q.: Optimal variational principle for backward stochastic control systems associated with Lévy processes, arXiv: 1010.4744v1. (2010)
Nualart, D., Schoutens, W.: BSDE’s and Feynman-Kac formula for Lévy process with application in finance. Bernoulli 7, 761–776 (2001)
Zhang, J., Ren, M., Tian, Y., Hou, G., Fang, F.: Constrained stochastic distribution control for nonlinear stochastic systems with non-Gaussian noises. Int. J. Innov. Comput. Inf. Control 9(4), 1759–1767 (2013)
Wang, H.Q., Chen, B., Lin, C.: Adaptive neural tracking control for a class of stochastic nonlinear systems with unknown dead-zone. Int. J. Innov. Comput. Inf. Control 9(8), 3257–3269 (2013)
Baghery, F., Øksendal, B.: A maximum principal for stochastic control with partial information. Stoch. Anal. Appl. 25, 493–514 (2007)
Kac, M.: Foundations of kinetic theory, In: Proc. 3-rd Berkeley Sympos. Math. Statist. Prob. 3, 171–197 (1956)
McKean, H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56, 1907–1911 (1966)
Hafayed, M., Abbas, S.: On near-optimal mean-field stochastic singular controls: necessary and sufficient conditions for near-optimality. J. Optim. Theory Appl. 160(3), 778–808 (2014)
Hafayed, M.: A mean-field necessary and sufficient conditions for optimal singular stochastic control. Commun. Math. Stat. 1, 417–435 (2014)
Hafayed, M.: A mean-field maximum principle for optimal control of forward–backward stochastic differential equations with Poisson jump processes. Int. J. Dyn. Control 1(4), 300–315 (2013)
Hafayed, M., Abba, A., Abbas, S.: On mean-field stochastic maximum principle for near-optimal controls for poisson jump diffusion with applications. Int. J. Dyn. Control 2, 262–284 (2014)
Hafayed, M.: Singular mean-field optimal control for forward-backward stochastic systems and applications to finance. Int. J. Dyn. Control 2(4), 542–554 (2014)
Buckdahn, R., Li, J., Peng, S.: Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Process. Appl. 119, 3133–3154 (2009)
Buckdahn, R., Djehiche, B., Li, J.: A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64, 197–216 (2011)
Shi, J.: Sufficient conditions of optimality for mean-field stochastic control problems. In: 12th International Conference on Control, Automation, Robotics & Vision Guangzhou, P.R. China December 5–7, pp. 747–752 (2012)
Ahmed, N.U.: Nonlinear diffusion governed by McKean–Vlasov equation on Hilbert space and optimal control. SIAM J. Control Optim. 46, 356–378 (2007)
Lasry, J.M., Lions, P.L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007)
Andersson, D., Djehiche, B.: A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63, 341–356 (2011)
Li, J.: Stochastic maximum principle in the mean-field controls. Automatica 48, 366–373 (2012)
Hafayed, M., Abbas, S.: A general maximum principle for stochastic differential equations of mean-field type with jump processes. Technical report, arXiv: 1301.7327v4. (2013)
Shen, Y., Siu, T.K.: The maximum principle for a jump-diffusion mean-field model and its application to the mean–variance problem. Nonlinear Anal. 86, 58–73 (2013)
Shen, Y., Meng, Q., Shi, P.: Maximum principle for mean-field jump-diffusions to stochastic delay differential equations and its applications to finance. Automatica 50, 1565–1579 (2014)
Yong, J.: A linear-quadratic optimal control problem for mean-field stochastic differential equations. SIAM J. Control Optim. 51(4), 2809–2838 (2013)
Wang, G., Zhang, C., Zhang, W.: Stochastic maximum principle for mean-field type optimal control under partial information. IEEE Trans. Automat. Control 59(2), 522–528 (2014)
Acknowledgments
The authors would like to thank the editor and anonymous referees for their constructive corrections and valuable suggestions that improved the manuscript considerably. The first author was supported by Algerian CNEPRU Project Grant B01420130137, 2014-2016.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hafayed, M., Abbas, S. & Abba, A. On Mean-Field Partial Information Maximum Principle of Optimal Control for Stochastic Systems with Lévy Processes. J Optim Theory Appl 167, 1051–1069 (2015). https://doi.org/10.1007/s10957-015-0762-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-015-0762-4
Keywords
- Optimal stochastic control
- Teugels martingales
- Mean-field stochastic differential equation
- Lévy processes
- Mean-field-type maximum principle
- Feedback control