VALUE IN MIXED STRATEGIES FOR ZERO-SUM STOCHASTIC DIFFERENTIAL GAMES WITHOUT ISAACS CONDITION

• 成果类型: Article
• 署名作者: Buckdahn, Rainer; Li, Juan; Quincampoix, Marc
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Bretagne Occidentale; Shandong University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP849
• 发表日期: 2014
• 页码: 1724-1768
• 关键词: viscosity solutions; EQUATIONS

摘要

In the present work, we consider 2-person zero-sum stochastic differential games with a nonlinear pay-off functional which is defined through a backward stochastic differential equation. Our main objective is to study for such a game the problem of the existence of a value without Isaacs condition. Not surprising, this requires a suitable concept of mixed strategies which, to the authors' best knowledge, was not known in the context of stochastic differential games. For this, we consider nonanticipative strategies with a delay defined through a partition pi of the time interval [0, T]. The underlying stochastic controls for the both players are randomized along pi by a hazard which is independent of the governing Brownian motion, and knowing the information available at the left time point t(j-i) of the subintervals generated by pi, the controls of Players 1 and 2 are conditionally independent over [t(j-1), t(j)). It is shown that the associated lower and upper value functions W-pi and U-pi converge uniformly on compacts to a function V, the so-called value in mixed strategies, as the mesh of pi tends to zero. This function V is characterized as the unique viscosity solution of the associated Hamilton Jacobi-Bellman-Isaacs equation.