Stochastic Differential Games and Viscosity Solutions of Hamilton--Jacobi--Bellman--Isaacs Equations
Stochastic Differential Games and Viscosity Solutions of Hamilton--Jacobi--Bellman--Isaacs Equations
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DOI：
10.1137/060671954
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发表时间：
2007-02
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影响因子：
--
通讯作者：
R. Buckdahn;Juan Li
中科院分区：
文献类型：
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作者：
R. Buckdahn;Juan Li
本文利用倒向随机微分方程理论研究了零和二人随机微分对策。更确切地说，我们概括了弗莱明和苏甘涅[印第安纳州大学数学J.，38（1989），pp. 293-314]通过考虑由受控BSDE定义的成本泛函，并通过允许容许的控制过程依赖于在游戏开始之前发生的事件。这类可容许控制过程的扩展的后果是成本泛函成为随机变量。然而，通过利用Girsanov变换参数，这是新的，在这种情况下，我们证明了游戏的上限和下限值函数仍然是确定性的。除了这类可容许控制过程的这种扩展是相当自然的并且反映了总是使用最大可用信息的参与者的行为这一事实之外，其与Bethesda方法的组合，特别是Peng [Bethesda and stochastic optimizations，in Topics in Stochastic Analysis，Science Press，Beijing，1997]引入的随机“后向半群”的概念，允许我们然后证明一个动态规划原则的上限和下限的价值函数的游戏在一个简单的方式。然后，上，下值函数是唯一的粘性解的上，下Hamilton-Jacobi-Bellman-Isaacs方程，分别。为此，将Peng的贝叶斯方法从随机控制理论的框架推广到随机微分对策的框架。
In this paper we study zero-sum two-player stochastic differential games with the help of the theory of backward stochastic differential equations (BSDEs). More precisely, we generalize the results of the pioneering work of Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp. 293-314] by considering cost functionals defined by controlled BSDEs and by allowing the admissible control processes to depend on events occurring before the beginning of the game. This extension of the class of admissible control processes has the consequence that the cost functionals become random variables. However, by making use of a Girsanov transformation argument, which is new in this context, we prove that the upper and the lower value functions of the game remain deterministic. Apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information, its combination with BSDE methods, in particular that of the notion of stochastic “backward semigroups" introduced by Peng [BSDE and stochastic optimizations, in Topics in Stochastic Analysis, Science Press, Beijing, 1997], allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way. The upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations, respectively. For this Peng's BSDE method is extended from the framework of stochastic control theory into that of stochastic differential games.