A Selective Survey of Recent Results on Linear-Quadratic Stochastic Differential Games

Abstract

This paper provides a brief survey of recent results on linear-quadratic stochastic differential games (LQ-SDGs), particularly LQ-SDGs in Nash and Stackelberg formulations. SDGs can be applied to study of various dynamic decision-making problems involving multiple players (or agents), where the players are subject to uncertainties presented in controlled stochastic differential equations (SDEs). In LQ-SDGs, the main mathematical techniques are stochastic maximum principles with forward-backward SDEs, dynamic programming principles with Hamilton-Jacobi partial differential equations, and four-step schemes with Riccati differential equations, by which explicit optimal solutions (saddle-point equilibrium in zero-sum SDGs, Nash equilibrium in nonzero-sum SDGs, and Stackelberg equilibrium in Stackelberg SDGs) can be identified under appropriate problem formulations. The paper first discusses various formulations of LQ-SDGs in Nash and Stackelberg settings. Then we state a number of their recent and important results. We also discuss LQ mean-field SDGs in Nash and Stackelberg formulations, which consider large-population SDGs, equivalently, large-scale macroscopic optimization decision-making problems. Finally, the conclusion of this paper identifies several important future directions of research in SDGs.