  The Hamilton-Jacobi-Bellman equation (HJB) associated with the time
inhomogeneous singular control problem is a parabolic partial differential
equation, and the existence of a classical solution is usually difficult to
prove. In this paper, a class of finite horizon stochastic singular control
problems of one dimensional diffusion is solved via a time inhomogeneous
zero-sum game (Dynkin game). The regularity of the value function of the Dynkin
game is investigated, and its integrated form coincides with the value function
of the singular control problem. We provide conditions under which a classical
solution to the associated HJB equation exists, thus the usual viscosity
solution approach is avoided. We also show that the optimal control policy is
to reflect the diffusion between two time inhomogeneous boundaries. For a more
general terminal payoff function, we showed that the optimal control involves a
possible impulse at maturity.
