Linear-quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Sun, Jingrui¹  Yong, Jiongmin¹ 

[1] Univ Cent Florida, Dept Math, Orlando, FL 32816 USA

关键词: Stochastic differential equation;    Linear-quadratic differential game;    Two-person;    Nonzero-sum;    Nash equilibrium;    Riccati differential equation;    Closed-loop;    Open-loop;   

DOI  :  10.1016/j.spa.2018.03.002

来源: Elsevier

【 摘 要 】

In this paper, we consider a linear-quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward-backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which could be different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of an open-loop saddle point coincides with that for the closed-loop saddle point, which leads to the conclusion that the closed-loop representation of an open-loop saddle point is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear- quadratic optimal control problem, the closed-loop representation of an open-loop optimal control coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist. (C) 2018 Elsevier B.V. All rights reserved.

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