# American Institute of Mathematical Sciences

December  2021, 11(4): 797-828. doi: 10.3934/mcrf.2020047

## A stackelberg game of backward stochastic differential equations with partial information

 School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author: Jingtao Shi

Received  December 2019 Revised  June 2020 Published  December 2021 Early access  November 2020

Fund Project: This work is financially supported by National Key R & D Program of China (2018YFB1305400) and National Natural Science Foundations of China (11971266, 11831010, 11571205)

This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs) with partial information, where the information of the follower is a sub-$\sigma$-algebra of that of the leader. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the partial information stochastic maximum principles of BSDEs and forward-backward stochastic differential equations (FBSDEs), respectively. Then a linear-quadratic (LQ) Stackelberg game of BSDEs with partial information is investigated. The state estimate feedback representation for the optimal control of the follower is first given via two Riccati equations. Then the leader's problem is formulated as an optimal control problem of FBSDE. Four high-dimensional Riccati equations are introduced to represent the state estimate feedback for the optimal control of the leader. Theoretic results are applied to a pension fund management problem of two players in the financial market.

Citation: Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2021, 11 (4) : 797-828. doi: 10.3934/mcrf.2020047
##### References:

show all references

##### References:
 [1] Weijun Meng, Jingtao Shi. A linear quadratic stochastic Stackelberg differential game with time delay. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021035 [2] Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026 [3] Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028 [4] Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022 [5] Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067 [6] Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97 [7] Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 [8] Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001 [9] Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021031 [10] Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261 [11] Russell Johnson, Carmen Núñez. Remarks on linear-quadratic dissipative control systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889 [12] Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100 [13] Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435 [14] Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035 [15] Yadong Shu, Bo Li. Linear-quadratic optimal control for discrete-time stochastic descriptor systems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021034 [16] Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117 [17] Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 [18] Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control & Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020 [19] Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401 [20] Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764

2020 Impact Factor: 1.284