不完全信息零和微分对策值的概率表示
A probabilistic representation for the value of zero-sum differential games with incomplete information on both sides
摘要：
证明了对于一类双方都不完全信息的零和微分对策，其值可以用概率表示为双方都控制连续鞅的完全信息零和随机微分对策的值。一个类似的表示作为一个控制问题的不连续鞅是已知的游戏与不完全信息的一方（见Cardaliaguet-Rainer [8]），我们的结果是一个连续时间模拟的所谓分裂游戏介绍在Laraki [20]和Sorin [27]为了分析离散时间模型。Cardaliaguet [4，5]证明了我们所考虑的对策的值是具有凸性约束的Hamilton-Jacobi方程的唯一解。因此，我们的结果提供了一个新的概率表示的Hamilton-Jacobi方程的凸性约束的值的随机微分对策具有无界控制空间和无界波动。
英文摘要：
We prove that for a class of zero-sum differential games with incomplete information on both sides, the value admits a probabilistic representation as the value of a zero-sum stochastic differential game with complete information, where both players control a continuous martingale. A similar representation as a control problem over discontinuous martingales was known for games with incomplete information on one side (see Cardaliaguet-Rainer [8]), and our result is a continuous-time analog of the so called splitting-game introduced in Laraki [20] and Sorin [27] in order to analyze discrete-time models. It was proved by Cardaliaguet [4, 5] that the value of the games we consider is the unique solution of some Hamilton-Jacobi equation with convexity constraints. Our result provides therefore a new probabilistic representation for solutions of Hamilton-Jacobi equations with convexity constraints as values of stochastic differential games with unbounded control spaces and unbounded volatility.
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