A Correction to "Dynkin Games Via Dirichlet Forms and Singular Control of One-Dimensional Diffusion"

In the paper "Dynkin Games Via Dirichlet Forms and Singular Control of One-Dimensional Diffusion", the authors tried to show the existences of a smooth value function and an optimal policy to a one-dimensional stochastic singular control problem, where the underlying process is a generalized diffusion process given by dX_t=\mu(X_t)dt+\sigma(X_t)dw_t, in which w_t is a Wiener process. It is found that either a condition on \mu(x) and \sigma(x) should be added, or a different diffusion process should be considered and as a result, the main theorem of this paper should be amended.