The Dynamic Programming Principle And Hamilton-Jaccobi-Bellman Equation Of The Stochastic System When It Is Constrained On A Domain

In this paper, we consider the stochastic system that constrained by a domain, and whose cost function is defined by Backward Stochastic Differential Equation, intro-duce the tool that Lions has used in1983to solve the generalized dynamic program-ming problem. We will discuss the continuous properties. The value function is shown to be continuous under some assumptions, and is the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation.