Before revision:  By identifying a controlled state process with reflection, we next transform the problem further into an equivalent auxiliary problem, which leads to a nonlinear Hamilton-Jacobi-Bellman (HJB)  with a Neumann boundary condition.
After revision:  By identifying a controlled state process with reflection, we next transform the problem further into an equivalent auxiliary problem, which leads to a nonlinear Hamilton-Jacobi-Bellman (HJB) equation with a Neumann boundary condition.