Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria
Linear–quadratic stochastic two-person nonzero-sum differential games: Open-loop and closed-loop Nash equilibria
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DOI：
10.1016/j.spa.2018.03.002
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发表时间：
2016-07
影响因子：
1.4
通讯作者：
Jingrui Sun;J. Yong
中科院分区：
文献类型：
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作者：
Jingrui Sun;J. Yong
在本文中，我们考虑线性二次随机两人非零和微分博弈。引入开环和闭环纳什均衡。前者的存在性表现为前向-后向随机微分方程组的可解性，后者的存在性表现为耦合对称Riccati微分方程组的可解性。有时，开环纳什均衡通过非对称 Riccati 方程组的解来承认闭环表示，这可能与一般闭环纳什均衡的结果不同。然而，我们发现，对于零和微分博弈的情况，开环鞍点的闭环表示的Riccati方程组与闭环鞍点的闭环表示是一致的，从而得出这样的结论：只要两者都存在，开环鞍点的闭环表示就是对应的闭环鞍点的结果。特别是，对于线性-二次最优控制问题，开环最优控制的闭环表示与相应闭环最优策略的结果一致，前提是两者都存在。
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.