A mean-field stochastic linear-quadratic optimal control problem with jumps under partial information

Yang, Yiyun; Tang, Maoning; Meng, Qingxin

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 53

In this article, the stochastic linear-quadratic optimal control problem of mean-field type with jumps under partial information is discussed. The state equation which contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson stochastic martingale measure, and the quadratic cost function contains cross terms. In addition, the state and the control as well as their expectations are contained both in the state equation and the cost functional. This is the so-called optimal control problem of mean-field type. Firstly, the existence and uniqueness of the optimal control is proved. Secondly, the adjoint processes of the state equation is introduced, and by using the duality technique, the optimal control is characterized by the stochastic Hamiltonian system. Thirdly, by applying a decoupling technology, we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information. Fourthly, the existence and uniqueness of the solutions of two Riccati equations are proved. Finally, we discuss a special case, and establish the corresponding feedback representation of the optimal control by means of filtering technique.

Classification : 93E20, 60H10
Keywords: Mean-field, linear-quadratic optimal control, partial information, Hamiltonian system, feedback representation, adjoint processes, Riccati equation

[1] A. Bensoussan, Stochastic control of partially observable systems. Cambridge University Press (1992). | MR | Zbl | DOI

[2] R. K. Boel and P. Varaiya, Optimal control of jump processes. SIAM J. Control Optim. 15 (1977) 92–119. | MR | Zbl | DOI

[3] R. Buckdahn, B. Djehiche and J. Li, A general stochastic maximum principle for SDEs of mean-field type. Appl. Math. Optim. 64 (2011) 197–216. | MR | Zbl | DOI

[4] I. Ekeland and R. Temam, Vol. 28 of Convex analysis and variational problems. SIAM, Philadelphia (1976). | MR | Zbl

[5] Y. Hu and B. Øksendal, Partial information linear quadratic control for jump diffusions. SIAM J. Control Optim. 47 (2008) 1744–1761. | MR | Zbl | DOI

[6] J. Huang, X. Li and J. Yong, A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Math. Control Related Fields 5 (2015) 97–139. | MR | Zbl | DOI

[7] P. Huang, G. Wang and H. Zhang, A partial information linear-quadratic optimal control problem of backward stochastic differential equation with its applications. Science 63 (2020) 188–200. | MR

[8] M. Kac, Foundations of kinetic theory. Third Berkeley Symposium on Mathematical Statistics and Probability (1956) 171–197. | MR | Zbl

[9] M. Kohlmann and X. Y. Zhou, Relationship between backward stochastic differential equations and stochastic controls: a linear-quadratic approach. SIAM J. Control Optim. 38 (2000) 1392–1392. | MR | Zbl | DOI

[10] X. Li, J. Sun and J. Xiong, Linear quadratic optimal control problems for mean-field backward stochastic differential equations. Appl. Math. Optim. 80 (2019) 223–250. | MR | Zbl | DOI

[11] A. E. B. Lim and X. Y. Zhou, Linear-quadratic control of backward stochastic differential equations. SIAM J. Control Optim. 40 (2001) 450–474. | MR | Zbl | DOI

[12] H. Ma and B. Liu, Linear-quadratic optimal control problem for partially observed forward-backward stochastic differential equations of mean-field type. Asian J. Control 18 (2016) 2146–2157. | MR | Zbl | DOI

[13] Q. Meng, General linear quadratic optimal stochastic control problem driven by a Brownian motion and a Poisson random martingale measure with random coefficients. Stoch. Anal. Appl. 32 (2014) 88–109. | MR | Zbl | DOI

[14] J. Moon and J. Chung, Indefinite linear-quadratic stochastic control problem for jump-diffusion models with random coefficients: a completion of squares approach. Mathematics 9 (2021) 2918. | DOI

[15] B. Øksendal and A. Sulem, Maximum principles for optimal control of forward-backward stochastic differential equations with jumps. SIAM J. Control Optim. 48 (2009) 2945–2976. | MR | Zbl | DOI

[16] J. Sun, Mean-field stochastic linear quadratic optimal control problems: open-loop solvabilities. ESAIM: COCV 23 (2017) 1099–1127. | MR | Zbl | Numdam

[17] J. Sun, X. Li and J. Yong, Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems. SIAM J. Control Optim. 54 (2016) 2274–2308. | MR | Zbl | DOI

[18] M. Tang and Q. Meng, Linear-quadratic optimal control problems for mean-field stochastic differential equations with jumps. Asian J. Control 21 (2019) 809–823. | MR | Zbl | DOI

[19] S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 (1994) 1447–1475. | MR | Zbl | DOI

[20] Z. Wu and X. Wang, FBSDE with Poisson process and its application to linear quadratic stochastic optimal control problem with random jumps. Acta Autom. Sin. 29 (2003) 821–826. | MR | Zbl

[21] J. Xiong, An introduction to stochastic filtering theory. Oxford University Press, London (2008). | MR | Zbl | DOI

[22] J. Yong, A linear-quadratic optimal control problem for mean-field stochastic differential equations. SIAM J. Control Optim. 51 (2011) 2809–2838. | MR | Zbl | DOI

[23] J. Yong and X. Y. Zhou, Stochastic controls: Hamiltonian systems and HJB equations. Springer-Verlag, New York (1999). | MR | Zbl | DOI

[24] F. Zhang, Y. Dong and Q. Meng, Backward stochastic Riccati equation with jumps associated with stochastic linear quadratic optimal control with jumps and random coefficients. SIAM J. Control Optim. 58 (2020) 393–424. | MR | Zbl | DOI

[25] H. Zhang, An optimal control problem for linear SDE of mean-field type with terminal constraint and partial information. Adv. Differ. Equ. 2019 (2019) 1–17. | MR | Zbl | DOI