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

Xun Li, Jingtao Shi, Jiongmin Yong

Research output: Journal article publicationJournal articleAcademic researchpeer-review

Abstract

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.

Original languageEnglish
Article number81
Pages (from-to)1-40
Number of pages40
JournalESAIM - Control, Optimisation and Calculus of Variations
Volume27
DOIs
Publication statusE-pub ahead of print - 23 Jul 2021

Keywords

  • Algebraic Riccati equations
  • Infinite horizon
  • MF-L2-stabilizability
  • Open-loop and closed-loop Nash equilibria
  • Static stabilizing solution
  • Two-person mean-field linear-quadratic stochastic differential game

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Control and Optimization
  • Computational Mathematics

Cite this