Weak Solution for a Class of Fully Nonlinear Stochastic Hamilton-Jacobi-Bellman Equations

This paper is concerned with the stochastic Hamilton-Jacobi-Bellman equation with controlled leading coefficients, which is a type of fully nonlinear backward stochastic partial differential equation (BSPDE for short). In order to formulate the weak solution for such kind of BSPDEs, the classical potential theory is generalized in the backward stochastic framework. The existence and uniqueness of the weak solution is proved, and for the partially non-Markovian case, we obtain the associated gradient estimate. As a byproduct, the existence and uniqueness of solution for a class of degenerate reflected BSPDEs is discussed as well.