最优控制理论与微分对策理论有着广泛的应用前景，如经济学，特别是金融问题以及生物、物理、工程、管理等科学领域。然而，目前国内外关于控制集无界的随机最优控制问题和微分对策问题的研究工作很少，本项目拟研究控制集无界的随机最优控制与随机微分对策问题。在控制集无界并且不要求相应的 Hamilton 函数具有凸性或凹性的前提下，利用动态规划方法拟研究：（i）随机最优控制问题相应的 Hamilton-Jacobi 方程的粘性解的存在唯一性；（ii）随机微分对策问题相应的上下值 Hamilton-Jacobi-Isaacs 方程的粘性解的存在唯一性，当 Isaacs 条件成立时，建立随机微分对策问题值函数的存在性；（iii）随机混合最优控制问题相应的 Hamilton-Jacobi 方程的粘性解的存在唯一性。同时举例说明本项目拟解决问题是十分尖锐的研究工作。

The optimal control theory and differential game theory have extensive application prospects, such as economics, especially financial issues as well as biology, physics, engineering, management and other fields of science. But there are rarely results on stochastic optimal controls and stochastic differential games with unbounded controls. This project intends to study the stochastic optimal controls problems and stochastic differential games with unbounded controls. We will study the following problems by using Dynamic Programming method: (i) The existence and uniqueness of the Hamilton-Jacobi equations which corresponding to the stochastic optimal control problem. (ii) The existence and uniqueness of the upper and lower Hamilton-Jacobi equations respectively, which corresponding to stochastic differential games. And to establish the existence of value function for the stochastic differential games under the Isaacs condition. (iii) The existence and uniqueness of the Hamilton-Jacobi equations which corresponding to stochastic combined optimal control problems. Then we will give some examples to show that the problems of this project are sharp in some sense.