论文标题
随机系数和随机汉密尔顿 - 雅各比 - 贝尔曼 - 伊萨克斯方程的随机差异游戏
Stochastic Differential Games with Random Coefficients and Stochastic Hamilton-Jacobi-Bellman-Isaacs Equations
论文作者
论文摘要
在本文中，我们研究了一类带有控制随机微分方程的零和两人随机差异游戏以及递归类型的回报/成本功能。与Fleming和Souganidis [Indiana Univ。数学。 J.，38（1989），pp。〜293--314]和Buckdahn和Li [Siam J. Control。我们首先证明了游戏的动态编程原理，然后在标准Lipschitz的连续性假设下，上下值函数被证明是上部和下部非线性非线性汉密尔顿 - 雅各尔顿 - 雅各布 - 雅各布 - 雅各布·贝尔曼 - 伊斯兰教徒（HJBI）方程的粘度解决方案。还证明了粘度解决方案的稳定性。在对扩散系数的某些其他规律性假设下，也解决了粘度解决方案的唯一性。
In this paper, we study a class of zero-sum two-player stochastic differential games with the controlled stochastic differential equations and the payoff/cost functionals of recursive type. As opposed to the pioneering work by Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp.~293--314] and the seminal work by Buckdahn and Li [SIAM J. Control Optim., 47 (2008), pp.~444--475], the involved coefficients may be random, going beyond the Markovian framework and leading to the random upper and lower value functions. We first prove the dynamic programming principle for the game, and then under the standard Lipschitz continuity assumptions on the coefficients, the upper and lower value functions are shown to be the viscosity solutions of the upper and the lower fully nonlinear stochastic Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. A stability property of viscosity solutions is also proved. Under certain additional regularity assumptions on the diffusion coefficient, the uniqueness of the viscosity solution is addressed as well.
