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<p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • Jun Moon,
  • Bing-Chang Wang,
  • Tamer Başar

摘要

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.