Skip to main content
SpringerLink
Log in

Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process

  • Original Paper
  • Published:
Communications on Applied Mathematics and Computation Aims and scope Submit manuscript

Abstract

This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels martingales associated with Lévy processes. In either case, we obtain the optimality system for the optimal controls in open-loop form, and by means of a decoupling technique, we obtain the optimal controls in closed-loop form which can be represented by two Riccati differential equations. Moreover, the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case.

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. Agram, N., Choutri, S.E.: Mean-field FBSDE and optimal control. Stoch. Anal. Appl. 39, 235–251 (2020)

    Article  MathSciNet  Google Scholar 

  2. Andersson, D., Djehiche, B.: A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63, 341–356 (2011)

    Article  MathSciNet  Google Scholar 

  3. Bahlali, K., Eddahbi, M., Essaky, E.: BSDE associated with L\(\acute{\text{e}}\)vy processes with application to PDIE. J. Appl. Math. Stoch. Anal. 16(1), 1–17 (2003)

    Article  Google Scholar 

  4. 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 

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

    Article  MathSciNet  Google Scholar 

  6. Chala, A.: The relaxed optimal control problem for mean-field SDEs systems and application. Automatic 50, 924–930 (2014)

    Article  MathSciNet  Google Scholar 

  7. Cramona, R., Delarue, F.: Mean field forward-backward stochastic differential equations. Electron. Commun. Probab. 18, 1–15 (2013)

    MathSciNet  Google Scholar 

  8. Deepa, R., Muthukumar, P.: Infinite horizon mean-field type relaxed optimal control with L\(\acute{\text{e}}\)vy processes. IFAC-PapersOnLine 51(1), 136–141 (2018)

    Article  Google Scholar 

  9. Di Nunno, G., Haferkorn, H.: A maximum principle for mean-field SDEs with time change. Appl. Math. Optim. 76, 137–176 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Huang, J., Li, X., Yong, J.: A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Math. Control Relat. Fields 5, 97–139 (2015)

    Article  MathSciNet  Google Scholar 

  12. Huang, J., Yu, Z.: Solvability of indefinite stochastic Riccati equations and linear quadratic optimal control problems. Syst. Control Lett. 68, 68–75 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Li, N., Li, X., Yu, Z.: Indefinite mean-field type linear-quadratic stochastic optimal control problems. Automatic 122, 1–28 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Li, N., Wu, Z., Yu, Z.: Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations. Sci. China Math. 61, 563–576 (2018)

    Article  MathSciNet  Google Scholar 

  16. Li, X., Sun, J., Yong, J.: Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability. Probab. Uncertainty Quant. Risk 1, 1–22 (2016)

    Article  MathSciNet  Google Scholar 

  17. Meng, Q., Tang, M.: Necessary and sufficient conditions for optimal control of stochastic systems associated with L\(\acute{{\rm e}}\)vy processes. Sci. China Ser. F Inf. Sci. 52(11), 1982–1992 (2009)

    Article  Google Scholar 

  18. Mitsui, K., Tabata, Y.: A stochastic linear-quadratic problem with L\(\acute{{\rm e}}\)vy processes and its application to finance. Stoch. Process Appl. 118, 120–152 (2008)

    Article  Google Scholar 

  19. Ni, Y., Zhang, J., Li, X.: Indefinite mean-field stochastic linear-quadratic optimal control. IEEE Trans. Autom. Control 60, 1786–1800 (2015)

    Article  MathSciNet  Google Scholar 

  20. Nualart, D., Schoutens, W.: Chaotic and predicatable representation for L\(\acute{{\rm e}}\)vy processes with applications in finance. Stoch. Process Appl. 90, 109–122 (2000)

    Article  Google Scholar 

  21. Nualart, D., Schoutens, W.: BSDE’s and Feynman-Kac formula for L\(\acute{{\rm e}}\)vy processes with applications in finance. Bernouli 7, 761–776 (2001)

    Article  Google Scholar 

  22. Sun, J., Wang, H.: Mean-field stochastic linear-quadratic optimal control problems: weak closed-loop solvability. arXiv:1907.01740 (2019)

  23. Tang, M., Meng, Q.: Linear-quadratic optimal control problems for mean-field stochastic differential equations with jumps. Asian J. Control 21, 809–823 (2019)

    Article  MathSciNet  Google Scholar 

  24. Tang, H., Wu, Z.: Stochastic differential equations and stochastic linear quadratic optimal control problem with L\(\acute{{\rm e}}\)vy processes. J. Syst. Sci. Complexity 22, 122–136 (2009)

    Article  MathSciNet  Google Scholar 

  25. Wei, Q., Yong, J., Yu, Z.: Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions. ESAIM Control Optim. Calc. Var. 25, 17–38 (2019)

    Article  MathSciNet  Google Scholar 

  26. Yong, J., Zhou, X.: Stochastic controls: Hamiltonian systems and HJB equations. In: Applications of Mathematics, vol. 43. Springer, New York (1999)

  27. Yong, J.: Linear-quadratic optimal control problems for mean-field stochastic differential equations. SIAM J. Control. Optim. 51, 2809–2838 (2013)

    Article  MathSciNet  Google Scholar 

  28. Yong, J.: Linear-quadratic stochastic optimal control problems for mean-field stochastic differential equations-time-consistent solutions. Trans. Am. Math. Soc. 369, 5467–5523 (2017)

    Article  MathSciNet  Google Scholar 

  29. Zong, G., Chen, Z.: Harnack inequality for mean-field stochastic differential equations. Stat. Probab. Lett. 83, 1424–1432 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank anonymous referees for helpful comments and suggestions which improved the original version of the paper. Q. Meng was supported by the Key Projects of Natural Science Foundation of Zhejiang Province of China (no. Z22A013952) and the National Natural Science Foundation of China (no. 11871121). Maoning Tang was supported by the Natural Science Foundation of Zhejiang Province of China (no. LY21A010001).

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Hong Xiong

  2. Department of Mathematical Sciences, Huzhou University, Huzhou, 313000, Zhejiang, China

    Maoning Tang & Qingxin Meng

Authors
  1. Hong Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maoning Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qingxin Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingxin Meng.

Ethics declarations

Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiong, H., Tang, M. & Meng, Q. Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process. Commun. Appl. Math. Comput. 4, 1386–1415 (2022). https://doi.org/10.1007/s42967-021-00181-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42967-021-00181-y

Keywords

Mathematics Subject Classification

Search

Navigation