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

doi:10.1007/s42967-021-00181-y

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (4): 1386-1415.doi: 10.1007/s42967-021-00181-y

Previous Articles Next Articles

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

Hong Xiong¹, Maoning Tang², Qingxin Meng²

1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China;
2. Department of Mathematical Sciences, Huzhou University, Huzhou, 313000, Zhejiang, China

Received:2021-08-26 Revised:2021-12-11 Online:2022-12-20 Published:2022-09-26
Supported by:
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).

Abstract

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.

Key words: Mean-field Teugels martingales, Linear-quadratic, Optimal control, Riccati equations, Feedback representation

CLC Number:

60H10
93E24

Hong Xiong, Maoning Tang, Qingxin Meng. Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1386-1415.

TrendMD

References

1. Agram, N., Choutri, S.E.:Mean-field FBSDE and optimal control. Stoch. Anal. Appl. 39, 235-251 (2020)
2. Andersson, D., Djehiche, B.:A maximum principle for SDEs of mean-field type. Appl. Math. Optim. 63, 341-356 (2011)
3. Bahlali, K., Eddahbi, M., Essaky, E.:BSDE associated with Lévy processes with application to PDIE. J. Appl. Math. Stoch. Anal. 16(1), 1-17 (2003)
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)
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)
6. Chala, A.:The relaxed optimal control problem for mean-field SDEs systems and application. Automatic 50, 924-930 (2014)
7. Cramona, R., Delarue, F.:Mean field forward-backward stochastic differential equations. Electron. Commun. Probab. 18, 1-15 (2013)
8. Deepa, R., Muthukumar, P.:Infinite horizon mean-field type relaxed optimal control with Lévy processes. IFAC-PapersOnLine 51(1), 136-141 (2018)
9. Di Nunno, G., Haferkorn, H.:A maximum principle for mean-field SDEs with time change. Appl. Math. Optim. 76, 137-176 (2017)
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)
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)
12. Huang, J., Yu, Z.:Solvability of indefinite stochastic Riccati equations and linear quadratic optimal control problems. Syst. Control Lett. 68, 68-75 (2014)
13. Li, J.:Stochastic maximum principle in the mean-field controls. Automatic 48, 366-373 (2012)
14. Li, N., Li, X., Yu, Z.:Indefinite mean-field type linear-quadratic stochastic optimal control problems. Automatic 122, 1-28 (2020)
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)
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)
17. Meng, Q., Tang, M.: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)
18. Mitsui, K., Tabata, Y.:A stochastic linear-quadratic problem with Lévy processes and its application to finance. Stoch. Process Appl. 118, 120-152 (2008)
19. Ni, Y., Zhang, J., Li, X.:Indefinite mean-field stochastic linear-quadratic optimal control. IEEE Trans. Autom. Control 60, 1786-1800 (2015)
20. Nualart, D., Schoutens, W.:Chaotic and predicatable representation for Lévy processes with applications in finance. Stoch. Process Appl. 90, 109-122 (2000)
21. Nualart, D., Schoutens, W.:BSDE's and Feynman-Kac formula for Lévy processes with applications in finance. Bernouli 7, 761-776 (2001)
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)
24. Tang, H., Wu, Z.:Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy processes. J. Syst. Sci. Complexity 22, 122-136 (2009)
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)
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)
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)
29. Zong, G., Chen, Z.:Harnack inequality for mean-field stochastic differential equations. Stat. Probab. Lett. 83, 1424-1432 (2013)

[1]	F. S. Mousavinejad, M. FatehiNia, A. Ebrahimi. P-Bifurcation of Stochastic van der Pol Model as a Dynamical System in Neuroscience [J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1293-1312.
[2]	Xiaoying Han, Habib N. Najm. Modeling Fast Diffusion Processes in Time Integration of Stiff Stochastic Differential Equations [J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1457-1493.

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

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 2

Recommended Articles

Metrics

Comments