TY - JOUR

T1 - Mean-field linear-quadratic stochastic differential games in an infinite horizon

AU - Li, Xun

AU - Shi, Jingtao

AU - Yong, Jiongmin

N1 - Funding Information:
∗This work was financially supported by Research Grants Council of Hong Kong under Grant 15213218 and 15215319, National Key R&D Program of China under Grant 2018YFB1305400, National Natural Science Funds of China under Grant 11971266, 11831010 and 11571205, China Scholarship Council, Shandong Provincial Natural Science Foundations under Grant ZR2020ZD24 and ZR2019ZD42, and NSF Grant DMS-1812921.
Publisher Copyright:
© The authors. Published by EDP Sciences, SMAI 2021.

PY - 2021/7/23

Y1 - 2021/7/23

N2 - This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. The existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.

AB - This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. The existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent.

KW - Algebraic Riccati equations

KW - Infinite horizon

KW - MF-L2-stabilizability

KW - Open-loop and closed-loop Nash equilibria

KW - Static stabilizing solution

KW - Two-person mean-field linear-quadratic stochastic differential game

UR - http://www.scopus.com/inward/record.url?scp=85111442885&partnerID=8YFLogxK

U2 - 10.1051/cocv/2021078

DO - 10.1051/cocv/2021078

M3 - Journal article

AN - SCOPUS:85111442885

VL - 27

SP - 1

EP - 40

JO - ESAIM - Control, Optimisation and Calculus of Variations

JF - ESAIM - Control, Optimisation and Calculus of Variations

SN - 1292-8119

M1 - 81

ER -