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

Finite Horizon Time Inhomogeneous Singular Control Problem of One-dimensional Diffusion via Dynkin Game