Skip to main content
SpringerLink
Log in

Stochastic Maximum Principle for Generalized Mean-Field Delay Control Problem

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we first derive the existence and uniqueness theorems for solutions to a class of generalized mean-field delay stochastic differential equations and mean-field anticipated backward stochastic differential equations (MFABSDEs). Then we study the stochastic maximum principle for generalized mean-field delay control problem. Since the state equation is distribution-depending, we define the adjoint equation as a MFABSDE in which all the derivatives of the coefficients are in Lions’ sense. We also provide a sufficient condition for the optimality of the control.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arkin, V., Saksonov, M.: Necessary optimality conditions for stochastic differential equations. Soviet Math. Doklady 20, 1–5 (1979)

    Google Scholar 

  2. Aurell, A., Djehiche, B.: Mean-field type modeling of nonlocal crowd aversion in pedestrian crowd dynamics. SIAM J. Control. Optim. 56(1), 434–455 (2018)

    Article  MathSciNet  Google Scholar 

  3. Aurell, A., Djehiche, B.: Modeling tagged pedestrian motion: a mean-field type game approach. Transp. Res. B Methodol. 121, 168–183 (2019)

    Article  Google Scholar 

  4. Bensoussan, A.: Lectures on stochastic control. In: Mitter, S. K., Moro, A. (eds.): Nonlinear Filtering and Stochastic Control, Lecture Notes in Mathematics, vol. 972, pp. 1-39. Springer-Verlag, Berlin (1982)

  5. Bismut, J.M.: An introductory approach to duality in optimal stochastic control. SIAM Rev. 20, 62–78 (1978)

    Article  MathSciNet  Google Scholar 

  6. Buckdahn, R., Djehiche, B., Li, J.: A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64, 197–216 (2011)

    Article  MathSciNet  Google Scholar 

  7. Buckdahn, R., Li, J., Ma, J.: A stochastic maximum principle for general mean-field systems. Appl. Math. Optim. 74, 507–534 (2016)

    Article  MathSciNet  Google Scholar 

  8. Buckdahn, R., Li, J., Peng, S.: Mean-field backward stochastic differential equations and related partial differential equations. Stoch. Proc. App. 119, 3133–3154 (2009)

    Article  MathSciNet  Google Scholar 

  9. Buckdahn, R., Li, J., Peng, S., Rainer, C.: Mean-field stochastic differential equations and associated PDEs. Ann. Probab. 45(2), 824–878 (2017)

    Article  MathSciNet  Google Scholar 

  10. Burger, M., Di Francesco, M., Markowich, P.A., Wolfram, M.-T.: Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete Continuous Dyn. Syst. Series B 19(5), 1311–1333 (2014)

    Article  MathSciNet  Google Scholar 

  11. Cardaliaguet, P.: Weak Solutions for First Order Mean Field Games with Local Coupling, pp. 111–158. In Analysis and Geometry in Control Theory and its Applications. Springer, Cham (2015)

    Google Scholar 

  12. Carmona, R.: Applications of mean field games in financial engineering and economic theory. arXiv:2012.05237 (2020)

  13. Carmona, R.: Lectures on BSDEs, stochastic control, and stochastic differential games with financial applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2016)

  14. Carmona, R., Delarue, F.: Probabilistic theory of mean field games with applications I, Probab. Theory Stoch. Model. vol. 83, Springer, Berlin (2018)

  15. Carmona, R., Delarue, F.: Probabilistic theory of mean field games with applications II, Probab. Theory Stoch. Model. vol. 84, Springer, Berlin (2018)

  16. Chen, L., Wu, Z.: Maximum principle for the stochastic optimal control problem with delay and application. Automatica 46(6), 1074–1080 (2010)

    Article  MathSciNet  Google Scholar 

  17. Djete, M., Possamaï, D., Tan, X.: McKean-Vlasov optimal control: the dynamic programming principle. Ann. Probab. 50(2), 791–833 (2022)

    Article  MathSciNet  Google Scholar 

  18. Du, H., Huang, J., Qin, Y.: A stochastic maximum principle for delayed mean-field stochastic differential equations and its applications. IEEE Trans. Auto. Control 58(12), 3212–3217 (2013)

    Article  MathSciNet  Google Scholar 

  19. Fouque, J.-P., Zhang, Z.: Mean field game with delay: a toy model. Risks 6(3), 90 (2018)

    Article  Google Scholar 

  20. Fouque, J.-P., Zhang, Z.: Deep learning methods for mean field control problems with delay. Front. Appl. Math. Stat. 6, 11 (2020)

    Article  Google Scholar 

  21. Göllmannn, L., Maurer, H.: Optimal control problems with time delays: Two case studies in biomedicine. Math. Biosci. Eng. 15(5), 1137–1154 (2018)

    Article  MathSciNet  Google Scholar 

  22. Haussmann, U.G.: A Stochastic Maximum Principle for Optimal Control of Diffusions, Essex. Longman Scientific and Technical, UK (1986)

    Google Scholar 

  23. Kushner, H.J.: On the stochastic maximum principle: fixed time of control. J. Math. Anal. Appl. 11, 78–92 (1965)

    Article  MathSciNet  Google Scholar 

  24. Kushner, H.J.: Necessary conditions for continuous parameter stochastic optimization problems. SIAM J. Control 10, 550–565 (1972)

    Article  MathSciNet  Google Scholar 

  25. Lachapelle, A., Wolfram, M.-T.: On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transp. Res. B: Methodol. 45, 1572–1589 (2011)

    Article  Google Scholar 

  26. Li, J.: Stochastic maximum principle in the mean-field controls. Automatica 48, 366–373 (2012)

    Article  MathSciNet  Google Scholar 

  27. Liu, L., Meng, X., Zhang, T.: Optimal control strategy for an impulsive stochastic competition system with time delays and jump. Phys. A 477, 99–113 (2017)

    Article  MathSciNet  Google Scholar 

  28. Ma, T., Meng, X., Chang, Z.: Dynamics and optimal harvesting control for a stochastic one-predator-two-prey time delay system with jumps. Complexity 19(2), 1–19 (2019)

    Google Scholar 

  29. McKean, H.P.: Propagation of chaos for a class of nonlinear parabolic equations. Lect. Series Differ. Equ. 7, 41–57 (1967)

    Google Scholar 

  30. Meng, Q., Shen, Y.: Optimal control of mean-field jump-diffusion systems with delay: A stochastic maximum principle approach. J. Comput. Appl. Math. 279, 13–30 (2015)

    Article  MathSciNet  Google Scholar 

  31. Nishio, K., Kashima, K., Imura, J.: Effects of time delay in feedback control of linear quantum systems. Phys. Rev. A 79(6), 062105 (2009)

    Article  Google Scholar 

  32. øksendal, B., Sulem, A., Zhang, T,: Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations. Adv. Appl. Probab. 43(2), 572–596 (2011)

  33. Peng, S.: A general stochastic maximum principle for optimal control problem. SIAM J. Control. Optim. 28(4), 966–979 (1990)

    Article  MathSciNet  Google Scholar 

  34. Peng, S., Yang, Z.: Anticipated backward stochastic differential equations. Ann. Probab. 37(3), 877–902 (2009)

    Article  MathSciNet  Google Scholar 

  35. Rigatos, G., Siano, P., Abbaszadeh, M., Ghosh, T.: Nonlinear optimal control of coupled time-delayed models of economic growth. Decisions Econ. Finan. 44, 375–399 (2021)

    Article  MathSciNet  Google Scholar 

  36. Shen, Y., Meng, Q., Shi, P.: Maximum principle for mean-field jump-diffusion stochastic delay differential equations and its application to finance. Automatica 50(6), 1565–1579 (2014)

    Article  MathSciNet  Google Scholar 

  37. Wang, S., James, M.R.: Quantum feedback control of linear stochastic systems with feedback-loop time delays. Automatica 52, 277–282 (2015)

    Article  MathSciNet  Google Scholar 

  38. Yong, J., Zhou, X.: Stochastic controls. Hamiltonian systems and HJB equations, Applications of Mathematics (New York), vol. 43, Springer Science & Business Media (1999)

Download references

Acknowledgements

The research of J. Xiong was supported by National Natural Science Foundation of China grants 61873325 and 11831010; the research of J.Y. Zheng was supported by National Natural Science Foundation of China grant 11901598 and Guangdong Characteristic Innovation Project No. 2023KTSCX163. The authors appreciate two anonymous referees for their careful review of the paper and the constructive feedback provided by the editors.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China

    Hancheng Guo

  2. Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China

    Jie Xiong

  3. Faculty of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, Shenzhen, 518172, China

    Jiayu Zheng

Authors
  1. Hancheng Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jie Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiayu Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiayu Zheng.

Additional information

Communicated by Bruno Bouchard.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, H., Xiong, J. & Zheng, J. Stochastic Maximum Principle for Generalized Mean-Field Delay Control Problem. J Optim Theory Appl 201, 352–377 (2024). https://doi.org/10.1007/s10957-024-02398-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-024-02398-2

Keywords

Mathematics Subject Classification

Search

Navigation