Some semismooth methods are considered to solve a nonsmooth equation which can arise from a discrete version of the well-known Hamilton-Jacobi-Bellman (HJB) equation, which is often encountered in optimal control and other applied areas. The authors first propose a semismooth Newton method and prove its monotone convergence by suitably choosing the initial iterative point and local superlinear convergence rate. Moreover, an inexact version of the proposed method is introduced, which reduces the cost of computations and still preserves nice convergence properties.